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## Precalculus

Mrs. Snow's Math

McNeil High School

Notes are intended to compliment the current text in use at McNeil High School Precalculus, Enhanced with Graphing Utilities, Texas Edition By Michael Sullivan and Michael Sullivan, III

## FALL SEMESTER

Fundamentals

MATHXL FOR SCHOOL – where to find it on line and where are all the assignments!

Algebra II Review Annotated Notes

Chapter 2:  Functions and Their Graphs All assignments are due at the start of the next class period.  The first assignments of this school year will be completed on paper:

Current assignments are currently not available on Mathxl, the precalculus online homework program.  For full credit, use separate paper:   please write your full name, class period, and identify the assignment with chapter.section number.  Write out each problem, show all work and circle the answer.

Lesson 2.1 and Lesson 2.2 Functions and 2.2 The Graph of a Function Mathxl Homework 2.1 and 2.2 Solutions 2.1 2.2 Mathxl -Textbook Homework 2.3 and 2.4 Solutions Textbook 2.3 and 2.4 Mathxl – Textbook Homework 2.5 Solutions 2.5 Annotated Notes Video Lesson 2.1 and 2.2 Lesson 2.3 and Lesson 2.4 Properties of Functions,   2.4  Piecewise and Greatest Integer Functions Annotated Notes 2.3 and 2.4 Video Lesson 2.3 and 2.4 Lesson 2.5   Graphing Techniques:  Transformations Annotated Notes 2.5 Video Lesson 2.5 In-class Transformation Practice Annotated Transformation Practice mrssnowsmath.com/…/chapter2review.2023r.pdf 2.5 Homework Worksheet Chapter 2 Review Review Solutions

Chapter 4:  Polynomial and Rational Functions

Lesson 4.1  Polynomial Functions and Model Annotated Notes Video Lesson 4.1 Lesson 4.2 The Real Zeros of a Polynomial Function Lesson 4.3 Complex Zeros; Fundamental Theorem of Algebra Annotated Notes 4.2 and 4.3 Video Lesson 4.2 and 4.3 Lesson 4.4 and Lesson 4.5  4.4 Properties of Rational Functions and.4.5 The Graphs of a Rational Function Annotated Notes 4.4 and 4.5 Video Lesson 4.4 and 4.5 Lesson 4.6 Polynomial and Rational Inequalities Annotated Notes 4.6 Video Lesson 4.6 Chapter 4 Spiral Ch 2 Review    Updated 9/2021 Spiral Review Solutions

Chapter 6 Trigonometric Functions

Quizlet Links: Sine, Cosine, and Tangent Radian Values Practice with radians from the unit circle; flashcards and more Unit Circle practice page   2 unit circles for drill practice

Lesson 6.1 and 6.2 Part 1 Angles and Their Measure, The Unit Circle Annotated Notes 6.1 And 6.2 Part 1 Video Lesson 6.1 and 6.2 part 1 Lesson 6.1 and 6.2 Part 2 Trigonometric Functions: Unit Circle Approach Annotated Notes 6.1 and 6-2 Part2 Video Lesson 6.1 and 6.2 part 2 2 Unit Circles   Practice Form Lesson 6.3 Properties of the Trigonometric Functions Annotated Notes Video Lesson 6.3 Review 6.1-6.3 Chapter 6.1-6.3 Review Solutions

Unit Circle Chart – In Order Lesson 6.4 Graphs of the Sine and Cosine Functions Annotated Notes 6.4 Video Lesson 6.4 Worksheet6.4 Lesson 6.5   Graphs of Secant and Cosecant Functions Annotated Notes 6.5 Part 1 Worksheet 6.5 Part 1   Graphing Secant and Cosecant Lesson 6.5 Graphs of Tangent and Cotangent Functions Annotated Notes 6.5 Part 2 Worksheet 6.5 Part 2   Graphing Tangent and Cotangent Video Lesson 6.5    this covers both part 1 and part 2:  Secant, Cosecant, Tangent and Cotangent Functions Lesson 6.6 Phase Shift Annotated Notes 6.6 Video Lesson 6.6 Worksheet 6.6 Lesson 8.5   Simple Harmonic Motion Annotated Notes 8.5 Video Lesson 8.5 Spiral Review 6.1-6.6, 8.5   Spiral Review Solutions

Lesson 5.1 and Lesson 5.2 Composite Functions and One-to-One Functions, Inverse Functions Annotated Notes 5.1 and 5.2 Video Lesson 5.1 and 5.2

Lesson 7.1 and 7.2   The Inverse Sine, Cosine, and Tangent Functions and The Inverse Trigonometric Function, Continued 7.1 and 7.2 Annotated Notes Video Lesson 7.1 Extra Examples Lesson 7.3 Trigonometric Equations Annotated Notes 7.3 Video Lesson 7.3 Review 5.1-5.2  and 7.1-7.3 Review 7.1-7.3 Solutions (revised 11/2/2019)

Unit Circle Sine, Cosine and Tangent values – Quizlet     link to a quizlet that has flash cards for our unit circle values.  The “Learn” link will quiz you with the flash cards out of order.

Trigonometric Identities Reference Sheet The first page will need to be memorized! Lesson 7.4 Trigonometric Identities Annotated Notes 7.4 Video Lesson 7.4 Worksheet Proofs Trig Identities   fall 2022 we are only doing the odd problems.  The even are available for additional practice Lesson 7.5 Sum and Difference Formulas Annotated Notes 7.5 Video Lesson 7.5 Lesson 7.6 Double-angle and Half-angle Formulas Annotated Notes 7.6 Video Lesson 7.6 Review Spiral 5.1-2, 7.1-3 7.4-7.5 Review Answers    Corrected!

Chapter 10

Lesson 10.1 and 10.2 Conics and the Parabola Annotated Notes 10.1 and 10.2 Video Lesson 10.1 and 10.2

Lesson 10.3 The Ellipse Annota te d Notes 10.3 Lesson 10.4 The Hyperbola Annotated Notes 10.4 Lesson 10.7 Plane Curves and Parametric Equations Annotated Notes 10.7

Fall Final Exam Review Fall Final Exam solutions

GUIDELINE FOR NOTECARD

This link will take you to an excel spreadsheet that will allow you to take your averages for either fall or spring semester and see what you need for the grading period or final to pass class.  This is designed for the grading cycles at McNeil High School. “WHAT IF…”GRADE CHECK  for Fall and Spring Semesters

## SPRING SEMESTER

Chapter 8 Lesson 8.1 Right Triangle Trigonometry; Applications Annotated Notes 8.1 Video 8.1 Trigonometry Applications Worksheet Lesson 8.2 The Law of Sines Lesson 8.2 The Ambiguous Case Example Annotated Notes 8.2 Video 8.2 Lesson 8.3 The Law of Cosines Lesson 8.4 Area of a Triangle Annotated Notes 8.3 and 8.4 Video 8.3 and 8.4 Review Chapter 8 Review Chapter  Solutions

Lesson 8.5 Simple Harmonic Motion (presented in the fall with Chapter 6) Annotated Notes 8.5 are located with Chapter 6

Chapter 9 Lesson 9.1 Polar Coordinates Annotated Notes 9.1 Video Lesson 9.1 Lesson 9.2 Polar Equations and Graphs Annotated Notes .9.2 Video Lesson 9.2 Limacon Examples Polar Graph Paper Lesson 9.4 Vectors Video Lesson 9.4 Annotated Notes 9.4 9.4 Static Equilibrium Problem #19 example Lesson 9.5 The Dot Product Annotated Notes 9.5 Video Lesson 9.5 Spiral Ch 8 and 9 Re view Spiral Review Solutions

Chapter 5 and 11.5 Lesson5.3 Exponential Functions Annotated Notes 5.3  and corrected Video Lesson 5.3 Lesson 5.4 Logarithmic Functions  and Lesson 5.5 Properties of Logarithms Annotated Notes 5.4 and 5.5 V ideo Lesson 5.4 Video Lesson 5.5

Lesson 5.6 Logarithmic and Exponential Equations Annotated Notes 5.6 Video Lesson 5.6 Lesson 5.7 and Lesson 5.8 Financial Models   and  Exponential Growth and Decay Models Annotated Notes 5.7 and 5.8 Video Lesson 5.7  Video Lesson 5.8 Chapter 5 Review  Review Solutions Chapter 5

Extra Lessons 11.5 and Rational Expressions Addition and Subtraction of Rational Expressions Annotated Notes Video Lesson Addition and Subtraction of Rational Expressions Multiplication and Division of Rational Expressions  Annotated Notes Video Lesson Multiplication and Division of Rational Expressions

Lesson 11.5 Partial Fraction Decomposition Annotated Notes 11.5 Video Lesson 11.5 https://www.youtube.com/watch?v=8lSCAY15BdQ

Chapter  12 Lesson 12.1 Sequences Annotated Notes 12.1 Video Lesson Lesson 12.2 Arithmetic Sequences Annotated Notes 12.2 Video Lesson Lesson 12.3   Geometric Sequences Lesson 12.3 Lecture Notes modified for voice over lecture Annotated Notes 12.3 Video Lesson Lesson 12.4 Mathematical Induction Lesson 12.4 Lecture Notes modified for voice over lecture Annotated Notes 12.4 Video Lesson 12.4 Homework Worksheet Lesson 12.5  The Binomial Theorem Lesson 12.5 Lecture Notes modified for voice over lecture Annotated Notes 12.5 Video Lesson Review Chapter 12

Solutions Review Ch 12 and 11.5   if you want credit for the review, make sure you show your work,  copying these answers will result in a 0%

Chapter 14 –  A Preview of Calculus

Lesson 14.1 Finding Limits Using Tables and Graphs Annotated Notes 14.1 *    Video Lecture 14.1 14.1 Worksheet

Lesson 14.2 Algebra Techniques for Finding Limits *    Video Lecture 14.2 Annotated Notes 14.2 14.2 Worksheet

Lesson 14.3  The Tangent Problem:  The Derivative Annotated Notes 14.3 *     Video Lecture 14.3 14.3 Worksheet Helpful videos for visualizing secant lines morphing into a tangent line: The Tangent Line and the Derivative   an 11 minute video, all is very good, the definition by derivative starts at 4 minutes. Applet:  Ordinary Derivative by Limit Definition   as an interactive applet to let the student see as the distance between points goes to zero,  the secant line becomes the tangent line.

Lesson 14.4  Limits at Infinity Annotated Notes 14.4 14.4 Worksheet *    Video Lecture 14.4

Lesson 14.5 The Area Problem:  The Integral Annotated Notes 14.5 14.5 Worksheet *    Video Lecture 14.5

Solutions to Chapter 14 practice problems

## 14.3, problem #7: correct answer is f'(x)=-5

Spiral Review Chapter 14 and Chapter 12 -remember, no work no credit!! Review Ch 12 and 14 Solutions

Note Card For Precalculus Final All students, whether exempting the final exam or not,  are required to complete the Final Exam Review.  Work must be shown for credit.   Spring Final Exam Review Spring Final Review Solutions

Other Notes

Intro to Calculus Lesson on Derivatives and Integration Annotated Notes Derivatives Worksheet #1 – Product and Quotient Rules Worksheet #2 – Chain Rule Chain Rule Notes Worksheet #3 – Integrals Integral Notes

This link will take you to an excel spreadsheet that will allow you to take your averages for either fall or spring semester and see what you need for the grading period or final to pass class.  This is designed for the grading cycles at McNeil High School. “WHAT IF…”GRADE CHECK for 2020-2021 School Year

## Module 7: Trigonometric Functions

The other trigonometric functions, learning outcomes.

• Find exact values of the trigonometric functions secant, cosecant, tangent, and cotangent of  30° (π/6),   45° (π/4), and 60° (π/3).
• Use reference angles to evaluate the trigonometric functions secant, cosecant, tangent, and cotangent.
• Use properties of even and odd trigonometric functions.
• Recognize and use fundamental identities.
• Evaluate trigonometric functions with a calculator.

## Find exact values of the trigonometric functions secant, cosecant, tangent, and cotangent

To define the remaining functions, we will once again draw a unit circle with a point $\left(x,y\right)$ corresponding to an angle of $t$, as shown in Figure 1. As with the sine and cosine, we can use the $\left(x,y\right)$ coordinates to find the other functions.

The first function we will define is the tangent. The tangent of an angle is the ratio of the y -value to the x -value of the corresponding point on the unit circle. In Figure 1, the tangent of angle $t$ is equal to $\frac{y}{x},x\ne 0$. Because the y -value is equal to the sine of $t$, and the x -value is equal to the cosine of $t$, the tangent of angle $t$ can also be defined as $\frac{\sin t}{\cos t},\cos t\ne 0$. The tangent function is abbreviated as $\tan$. The remaining three functions can all be expressed as reciprocals of functions we have already defined.

• The secant function is the reciprocal of the cosine function. In Figure 1, the secant of angle $t$ is equal to $\frac{1}{\cos t}=\frac{1}{x},x\ne 0$. The secant function is abbreviated as $\sec$.
• The cotangent function is the reciprocal of the tangent function. In Figure 1, the cotangent of angle $t$ is equal to $\frac{\cos t}{\sin t}=\frac{x}{y},y\ne 0$. The cotangent function is abbreviated as $\cot$.
• The cosecant function is the reciprocal of the sine function. In Figure 1, the cosecant of angle $t$ is equal to $\frac{1}{\sin t}=\frac{1}{y},y\ne 0$. The cosecant function is abbreviated as $\csc$.

## A General Note: Tangent, Secant, Cosecant, and Cotangent Functions

If $t$ is a real number and $\left(x,y\right)$ is a point where the terminal side of an angle of $t$ radians intercepts the unit circle, then

$\begin{gathered}\tan t=\frac{y}{x},x\ne 0\\ \sec t=\frac{1}{x},x\ne 0\\ \csc t=\frac{1}{y},y\ne 0\\ \cot t=\frac{x}{y},y\ne 0\end{gathered}$

## Example 1: Finding Trigonometric Functions from a Point on the Unit Circle

The point $\left(-\frac{\sqrt{3}}{2},\frac{1}{2}\right)$ is on the unit circle, as shown in Figure 2. Find $\sin t,\cos t,\tan t,\sec t,\csc t$, and $\cot t$.

Because we know the $\left(x,y\right)$ coordinates of the point on the unit circle indicated by angle $t$, we can use those coordinates to find the six functions:

$\begin{gathered}\sin t=y=\frac{1}{2}\\ \cos t=x=-\frac{\sqrt{3}}{2}\\ \tan t=\frac{y}{x}=\frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}}=\frac{1}{2}\left(-\frac{2}{\sqrt{3}}\right)=-\frac{1}{\sqrt{3}}=-\frac{\sqrt{3}}{3}\\ \sec t=\frac{1}{x}=\frac{1}{\frac{-\frac{\sqrt{3}}{2}}{}}=-\frac{2}{\sqrt{3}}=-\frac{2\sqrt{3}}{3}\\ \csc t=\frac{1}{y}=\frac{1}{\frac{1}{2}}=2\\ \cot t=\frac{x}{y}=\frac{-\frac{\sqrt{3}}{2}}{\frac{1}{2}}=-\frac{\sqrt{3}}{2}\left(\frac{2}{1}\right)=-\sqrt{3}\end{gathered}$

The point $\left(\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\right)$ is on the unit circle, as shown in Figure 3. Find $\sin t,\cos t,\tan t,\sec t,\csc t$, and $\cot t$.

$\sin t=-\frac{\sqrt{2}}{2},\cos t=\frac{\sqrt{2}}{2},\tan t=-1,\sec t=\sqrt{2},\csc t=-\sqrt{2},\cot t=-1$

## Example 2: Finding the Trigonometric Functions of an Angle

Find $\sin t,\cos t,\tan t,\sec t,\csc t$, and $\cot t$ when $t=\frac{\pi }{6}$.

We have previously used the properties of equilateral triangles to demonstrate that $\sin \frac{\pi }{6}=\frac{1}{2}$ and $\cos \frac{\pi }{6}=\frac{\sqrt{3}}{2}$. We can use these values and the definitions of tangent, secant, cosecant, and cotangent as functions of sine and cosine to find the remaining function values.

$\begin{gathered} \tan \frac{\pi }{6}=\frac{\sin\frac{\pi }{6}}{\cos \frac{\pi }{6}} =\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}\end{gathered}$

$\begin{gathered}\sec \frac{\pi }{6}=\frac{1}{\cos \frac{\pi }{6}} =\frac{1}{\frac{\sqrt{3}}{2}}=\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}\end{gathered}$

$\begin{gathered}\csc \frac{\pi }{6}=\frac{1}{\sin \frac{\pi }{6}}=\frac{1}{\frac{1}{2}}=2\end{gathered}$

$\begin{gathered}\cot \frac{\pi }{6}=\frac{\cos \frac{\pi }{6}}{\sin \frac{\pi }{6}} =\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}=\sqrt{3} \end{gathered}$

Find $\sin t,\cos t,\tan t,\sec t,\csc t$, and $\cot t$ when $t=\frac{\pi }{3}$.

\begin{align}&\sin \frac{\pi }{3}=\frac{\sqrt{3}}{2}\\ &\cos \frac{\pi }{3}=\frac{1}{2}\\ &\tan \frac{\pi }{3}=\sqrt{3}\\ &\sec \frac{\pi }{3}=2\\ &\csc \frac{\pi }{3}=\frac{2\sqrt{3}}{3}\\ &\cot \frac{\pi }{3}=\frac{\sqrt{3}}{3}\end{align}

Because we know the sine and cosine values for the common first-quadrant angles, we can find the other function values for those angles as well by setting $x$ equal to the cosine and $y$ equal to the sine and then using the definitions of tangent, secant, cosecant, and cotangent. The results are shown in the table below.

## Using Reference Angles to Evaluate Tangent, Secant, Cosecant, and Cotangent

We can evaluate trigonometric functions of angles outside the first quadrant using reference angles as we have already done with the sine and cosine functions. The procedure is the same: Find the reference angle formed by the terminal side of the given angle with the horizontal axis. The trigonometric function values for the original angle will be the same as those for the reference angle, except for the positive or negative sign, which is determined by x – and y -values in the original quadrant. Figure 4 shows which functions are positive in which quadrant.

To help us remember which of the six trigonometric functions are positive in each quadrant, we can use the mnemonic phrase “A Smart Trig Class.” Each of the four words in the phrase corresponds to one of the four quadrants, starting with quadrant I and rotating counterclockwise. In quadrant I, which is “ A ,” a ll of the six trigonometric functions are positive. In quadrant II, “ S mart,” only s ine and its reciprocal function, cosecant, are positive. In quadrant III, “ T rig,” only t angent and its reciprocal function, cotangent, are positive. Finally, in quadrant IV, “ C lass,” only c osine and its reciprocal function, secant, are positive.

## How To: Given an angle not in the first quadrant, use reference angles to find all six trigonometric functions.

• Measure the angle formed by the terminal side of the given angle and the horizontal axis. This is the reference angle.
• Evaluate the function at the reference angle.
• Observe the quadrant where the terminal side of the original angle is located. Based on the quadrant, determine whether the output is positive or negative.

## Example 3: Using Reference Angles to Find Trigonometric Functions

Use reference angles to find all six trigonometric functions of $-\frac{5\pi }{6}$.

The angle between this angle’s terminal side and the x -axis is $\frac{\pi }{6}$, so that is the reference angle. Since $-\frac{5\pi }{6}$ is in the third quadrant, where both $x$ and $y$ are negative, cosine, sine, secant, and cosecant will be negative, while tangent and cotangent will be positive.

$\begin{gathered}\cos \left(-\frac{5\pi }{6}\right)=-\frac{\sqrt{3}}{2} \\ \sin \left(-\frac{5\pi }{6}\right)=-\frac{1}{2} \\ \tan\left(-\frac{5\pi }{6}\right)=\frac{\sqrt{3}}{3} \\ \sec\left(-\frac{5\pi }{6}\right)=-\frac{2\sqrt{3}}{3}\\ \csc\left(-\frac{5\pi }{6}\right)=-2\\ \cot \left(-\frac{5\pi }{6}\right)=\sqrt{3} \end{gathered}$

Use reference angles to find all six trigonometric functions of $-\frac{7\pi }{4}$.

$\sin \left(-\frac{7\pi }{4}\right)=\frac{\sqrt{2}}{2},\cos \left(-\frac{7\pi }{4}\right)=\frac{\sqrt{2}}{2},\tan \left(-\frac{7\pi }{4}\right)=1$, $\sec \left(-\frac{7\pi }{4}\right)=\sqrt{2},\csc \left(-\frac{7\pi }{4}\right)=\sqrt{2},\cot \left(-\frac{7\pi }{4}\right)=1$

## Using Even and Odd Trigonometric Functions

To be able to use our six trigonometric functions freely with both positive and negative angle inputs, we should examine how each function treats a negative input. As it turns out, there is an important difference among the functions in this regard.

Consider the function $f\left(x\right)={x}^{2}$, shown in Figure 5. The graph of the function is symmetrical about the y -axis. All along the curve, any two points with opposite x -values have the same function value. This matches the result of calculation: ${\left(4\right)}^{2}={\left(-4\right)}^{2}$, ${\left(-5\right)}^{2}={\left(5\right)}^{2}$, and so on. So $f\left(x\right)={x}^{2}$ is an even function , a function such that two inputs that are opposites have the same output. That means $f\left(-x\right)=f\left(x\right)$.

Figure 5.  The function $f\left(x\right)={x}^{2}$ is an even function.

Now consider the function $f\left(x\right)={x}^{3}$, shown in Figure 6. The graph is not symmetrical about the y -axis. All along the graph, any two points with opposite x -values also have opposite y -values. So $f\left(x\right)={x}^{3}$ is an odd function , one such that two inputs that are opposites have outputs that are also opposites. That means $f\left(-x\right)=-f\left(x\right)$.

Figure 6.  The function $f\left(x\right)={x}^{3}$ is an odd function.

We can test whether a trigonometric function is even or odd by drawing a unit circle with a positive and a negative angle, as in Figure 7. The sine of the positive angle is $y$. The sine of the negative angle is − y . The sine function , then, is an odd function. We can test each of the six trigonometric functions in this fashion. The results are shown in in the table below.

## A General Note: Even and Odd Trigonometric Functions

An even function is one in which $f\left(-x\right)=f\left(x\right)$.

An odd function is one in which $f\left(-x\right)=-f\left(x\right)$.

Cosine and secant are even:

$\begin{gathered}\cos \left(-t\right)=\cos t \\ \sec \left(-t\right)=\sec t \end{gathered}$

Sine, tangent, cosecant, and cotangent are odd:

$\begin{gathered}\sin \left(-t\right)=-\sin t \\ \tan \left(-t\right)=-\tan t \\ \csc \left(-t\right)=-\csc t \\ \cot \left(-t\right)=-\cot t \end{gathered}$

## Example 4: Using Even and Odd Properties of Trigonometric Functions

If the $\sec t=2$, what is the $\sec (-t)$?

Secant is an even function. The secant of an angle is the same as the secant of its opposite. So if the secant of angle t is 2, the secant of $-t$ is also 2.

If the $\cot t=\sqrt{3}$, what is $\cot (-t)$?

$-\sqrt{3}$

## Recognize and Use Fundamental Identities

We have explored a number of properties of trigonometric functions. Now, we can take the relationships a step further, and derive some fundamental identities. Identities are statements that are true for all values of the input on which they are defined. Usually, identities can be derived from definitions and relationships we already know. For example, the Pythagorean Identity we learned earlier was derived from the Pythagorean Theorem and the definitions of sine and cosine.

## A General Note: Fundamental Identities

We can derive some useful identities from the six trigonometric functions. The other four trigonometric functions can be related back to the sine and cosine functions using these basic relationships:

$\tan t=\frac{\sin t}{\cos t}$

$\sec t=\frac{1}{\cos t}$

$\csc t=\frac{1}{\sin t}$

$\cot t=\frac{1}{\tan t}=\frac{\cos t}{\sin t}$

## Example 5: Using Identities to Evaluate Trigonometric Functions

• Given $\sin \left(45^\circ \right)=\frac{\sqrt{2}}{2},\cos \left(45^\circ \right)=\frac{\sqrt{2}}{2}$, evaluate $\tan \left(45^\circ \right)$.
• Given $\sin \left(\frac{5\pi }{6}\right)=\frac{1}{2},\cos\left(\frac{5\pi }{6}\right)=-\frac{\sqrt{3}}{2}$, evaluate $\sec \left(\frac{5\pi }{6}\right)$.

Because we know the sine and cosine values for these angles, we can use identities to evaluate the other functions.

• \begin{align}\tan \left(45^\circ \right)=\frac{\sin \left(45^\circ \right)}{\cos \left(45^\circ \right)} =\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} =1 \end{align}
• \begin{align}\sec \left(\frac{5\pi }{6}\right)=\frac{1}{\cos \left(\frac{5\pi }{6}\right)} =\frac{1}{-\frac{\sqrt{3}}{2}} =\frac{-2\sqrt{3}}{1} =\frac{-2}{\sqrt{3}} =-\frac{2\sqrt{3}}{3} \end{align}

Evaluate $\csc\left(\frac{7\pi }{6}\right)$.

$-2$

## Example 6: Using Identities to Simplify Trigonometric Expressions

Simplify $\frac{\sec t}{\tan t}$.

We can simplify this by rewriting both functions in terms of sine and cosine.

\begin{align} \frac{\sec t}{\tan t}&=\frac{\frac{1}{\cos t}}{\frac{\sin t}{\cos t}}&& \text{To divide the functions, we multiply by the reciprocal.} \\&=\frac{1}{\cos t}\frac{\cos t}{\sin t}&&\text{Divide out the cosines.} \\ &=\frac{1}{\sin t}&&\text{Simplify and use the identity.}\\ &=\csc t \end{align}

By showing that $\frac{\sec t}{\tan t}$ can be simplified to $\csc t$, we have, in fact, established a new identity.

$\frac{\sec t}{\tan t}=\csc t$

Simplify $\tan t\left(\cos t\right)$.

$\sin t$

## Alternate Forms of the Pythagorean Identity

We can use these fundamental identities to derive alternative forms of the Pythagorean Identity , ${\cos }^{2}t+{\sin }^{2}t=1$. One form is obtained by dividing both sides by ${\cos }^{2}t:$

The other form is obtained by dividing both sides by ${\sin }^{2}t:$

## A General Note: Alternate Forms of the Pythagorean Identity

$1+{\tan }^{2}t={\sec }^{2}t$

${\cot }^{2}t+1={\csc }^{2}t$

## Example 7: Using Identities to Relate Trigonometric Functions

If $\text{cos}\left(t\right)=\frac{12}{13}$ and $t$ is in quadrant IV, as shown in Figure 8, find the values of the other five trigonometric functions.

We can find the sine using the Pythagorean Identity, ${\cos }^{2}t+{\sin }^{2}t=1$, and the remaining functions by relating them to sine and cosine.

$\begin{gathered}{\left(\frac{12}{13}\right)}^{2}+{\sin }^{2}t=1 \\ {\sin }^{2}t=1-{\left(\frac{12}{13}\right)}^{2} \\ {\sin }^{2}t=1-\frac{144}{169} \\ {\sin }^{2}t=\frac{25}{169} \\ \sin t=\pm \sqrt{\frac{25}{169}} \\ \sin t=\pm \frac{\sqrt{25}}{\sqrt{169}} \\ \sin t=\pm \frac{5}{13} \end{gathered}$

The sign of the sine depends on the y -values in the quadrant where the angle is located. Since the angle is in quadrant IV, where the y -values are negative, its sine is negative, $-\frac{5}{13}$.

The remaining functions can be calculated using identities relating them to sine and cosine.

$\begin{gathered}\tan t=\frac{\sin t}{\cos t}=\frac{-\frac{5}{13}}{\frac{12}{13}}=-\frac{5}{12} \\ \sec t=\frac{1}{\cos t}=\frac{1}{\frac{12}{13}}=\frac{13}{12} \\ \csc t=\frac{1}{\sin t}=\frac{1}{-\frac{5}{13}}=-\frac{13}{5}\\ \cot t=\frac{1}{\tan t}=\frac{1}{-\frac{5}{12}}=-\frac{12}{5}\end{gathered}$

If $\sec \left(t\right)=-\frac{17}{8}$ and $0<t<\pi$, find the values of the other five functions.

$\cos t=-\frac{8}{17},\sin t=\frac{15}{17},\tan t=-\frac{15}{8},\csc t=\frac{17}{15},\cot t=-\frac{8}{15}$

As we discussed in the chapter opening, a function that repeats its values in regular intervals is known as a periodic function . The trigonometric functions are periodic. For the four trigonometric functions, sine, cosine, cosecant and secant, a revolution of one circle, or $2\pi$, will result in the same outputs for these functions. And for tangent and cotangent, only a half a revolution will result in the same outputs.

Other functions can also be periodic. For example, the lengths of months repeat every four years. If $x$ represents the length time, measured in years, and $f\left(x\right)$ represents the number of days in February, then $f\left(x+4\right)=f\left(x\right)$. This pattern repeats over and over through time. In other words, every four years, February is guaranteed to have the same number of days as it did 4 years earlier. The positive number 4 is the smallest positive number that satisfies this condition and is called the period. A period is the shortest interval over which a function completes one full cycle—in this example, the period is 4 and represents the time it takes for us to be certain February has the same number of days.

## A General Note: Period of a Function

The period $P$ of a repeating function $f$ is the number representing the interval such that $f\left(x+P\right)=f\left(x\right)$ for any value of $x$.

The period of the cosine, sine, secant, and cosecant functions is $2\pi$.

The period of the tangent and cotangent functions is $\pi$.

## Example 8: Finding the Values of Trigonometric Functions

Find the values of the six trigonometric functions of angle $t$ based on Figure 9 .

$\begin{gathered}\sin t=y=-\frac{\sqrt{3}}{2}\\ \cos t=x=-\frac{1}{2}\\ \tan t=\frac{\sin t}{\cos t}=\frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}}=\sqrt{3}\\ \sec t=\frac{1}{\cos t}=\frac{1}{-\frac{1}{2}}=-2\\ \csc t=\frac{1}{\sin t}=\frac{1}{-\frac{\sqrt{3}}{2}}=-\frac{2\sqrt{3}}{3}\\ \cot t=\frac{1}{\tan t}=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}\end{gathered}$

Find the values of the six trigonometric functions of angle $t$ based on Figure 10.

\begin{align}&\sin t=-1\\&\cos t=0\\&\tan t \text{ is undefined}\\ &\sec t \text{ is undefined}\\&\csc t=-1\\&\cot t=0\end{align}

## Example 9: Finding the Value of Trigonometric Functions

If $\sin \left(t\right)=-\frac{\sqrt{3}}{2}$ and $\text{cos}\left(t\right)=\frac{1}{2}$, find $\text{sec}\left(t\right),\text{csc}\left(t\right),\text{tan}\left(t\right),\text{ cot}\left(t\right)$.

$\begin{gathered} \sec t=\frac{1}{\cos t}=\frac{1}{\frac{1}{2}}=2 \\ \csc t=\frac{1}{\sin t}=\frac{1}{-\frac{\sqrt{3}}{2}}-\frac{2\sqrt{3}}{3} \\ \tan t=\frac{\sin t}{\cos t}=\frac{-\frac{\sqrt{3}}{2}}{\frac{1}{2}}=-\sqrt{3} \\ \cot t=\frac{1}{\tan t}=\frac{1}{-\sqrt{3}}=-\frac{\sqrt{3}}{3} \end{gathered}$

If $\sin \left(t\right)=\frac{\sqrt{2}}{2}$ and $\cos \left(t\right)=\frac{\sqrt{2}}{2}$, find $\text{sec}\left(t\right),\text{csc}\left(t\right),\text{tan}\left(t\right),\text{ and cot}\left(t\right)$.

$\sec t=\sqrt{2},\csc t=\sqrt{2},\tan t=1,\cot t=1$

## Evaluating Trigonometric Functions with a Calculator

We have learned how to evaluate the six trigonometric functions for the common first-quadrant angles and to use them as reference angles for angles in other quadrants. To evaluate trigonometric functions of other angles, we use a scientific or graphing calculator or computer software. If the calculator has a degree mode and a radian mode, confirm the correct mode is chosen before making a calculation.

Evaluating a tangent function with a scientific calculator as opposed to a graphing calculator or computer algebra system is like evaluating a sine or cosine: Enter the value and press the TAN key. For the reciprocal functions, there may not be any dedicated keys that say CSC, SEC, or COT. In that case, the function must be evaluated as the reciprocal of a sine, cosine, or tangent.

If we need to work with degrees and our calculator or software does not have a degree mode, we can enter the degrees multiplied by the conversion factor $\frac{\pi }{180}$ to convert the degrees to radians. To find the secant of $30^\circ$, we could press

## How To: Given an angle measure in radians, use a scientific calculator to find the cosecant.

• If the calculator has degree mode and radian mode, set it to radian mode.
• Enter: $1\text{ /}$
• Enter the value of the angle inside parentheses.
• Press the SIN key.
• Press the = key.

## How To: Given an angle measure in radians, use a graphing utility/calculator to find the cosecant.

• If the graphing utility has degree mode and radian mode, set it to radian mode.
• Press the ENTER key.

## Example 10: Evaluating the Secant Using Technology

Evaluate the cosecant of $\frac{5\pi }{7}$.

For a scientific calculator, enter information as follows:

$\text{1 / ( 5 }\times \text{ }\pi \text{ / 7 ) SIN =}$

$\csc \left(\frac{5\pi }{7}\right)\approx 1.279$

Evaluate the cotangent of $-\frac{\pi }{8}$.

$\approx -2.414$

## Key Equations

Key concepts.

• The tangent of an angle is the ratio of the y -value to the x -value of the corresponding point on the unit circle.
• The secant, cotangent, and cosecant are all reciprocals of other functions. The secant is the reciprocal of the cosine function, the cotangent is the reciprocal of the tangent function, and the cosecant is the reciprocal of the sine function.
• The six trigonometric functions can be found from a point on the unit circle.
• Trigonometric functions can also be found from an angle.
• Trigonometric functions of angles outside the first quadrant can be determined using reference angles.
• A function is said to be even if $f\left(-x\right)=f\left(x\right)$ and odd if $f\left(-x\right)=-f\left(x\right)$.
• Cosine and secant are even; sine, tangent, cosecant, and cotangent are odd.
• Even and odd properties can be used to evaluate trigonometric functions.
• The Pythagorean Identity makes it possible to find a cosine from a sine or a sine from a cosine.
• Identities can be used to evaluate trigonometric functions.
• Fundamental identities such as the Pythagorean Identity can be manipulated algebraically to produce new identities.
• The trigonometric functions repeat at regular intervals.
• The period $P$ of a repeating function $f$ is the smallest interval such that $f\left(x+P\right)=f\left(x\right)$ for any value of $x$.
• The values of trigonometric functions of special angles can be found by mathematical analysis.
• To evaluate trigonometric functions of other angles, we can use a calculator or computer software.
• Precalculus. Authored by : OpenStax College. Provided by : OpenStax. Located at : http://cnx.org/contents/fd53eae1-fa23-47c7-[email protected]:1/Preface . License : CC BY: Attribution
• Determining Trigonometric Function Values on the Calculator . Authored by : Mathispower4u. Located at : https://youtu.be/rhRi_IuE_18 . License : All Rights Reserved

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• Introduction to Functions
• 1.1 Functions and Function Notation
• 1.2 Domain and Range
• 1.3 Rates of Change and Behavior of Graphs
• 1.4 Composition of Functions
• 1.5 Transformation of Functions
• 1.6 Absolute Value Functions
• 1.7 Inverse Functions
• Key Equations
• Key Concepts

## Review Exercises

Practice test.

• Introduction to Linear Functions
• 2.1 Linear Functions
• 2.2 Graphs of Linear Functions
• 2.3 Modeling with Linear Functions
• 2.4 Fitting Linear Models to Data
• Introduction to Polynomial and Rational Functions
• 3.1 Complex Numbers
• 3.3 Power Functions and Polynomial Functions
• 3.4 Graphs of Polynomial Functions
• 3.5 Dividing Polynomials
• 3.6 Zeros of Polynomial Functions
• 3.7 Rational Functions
• 3.8 Inverses and Radical Functions
• 3.9 Modeling Using Variation
• Introduction to Exponential and Logarithmic Functions
• 4.1 Exponential Functions
• 4.2 Graphs of Exponential Functions
• 4.3 Logarithmic Functions
• 4.4 Graphs of Logarithmic Functions
• 4.5 Logarithmic Properties
• 4.6 Exponential and Logarithmic Equations
• 4.7 Exponential and Logarithmic Models
• 4.8 Fitting Exponential Models to Data
• Introduction to Trigonometric Functions

## 5.2 Unit Circle: Sine and Cosine Functions

5.3 the other trigonometric functions, 5.4 right triangle trigonometry.

• Introduction to Periodic Functions
• 6.1 Graphs of the Sine and Cosine Functions
• 6.2 Graphs of the Other Trigonometric Functions
• 6.3 Inverse Trigonometric Functions
• Introduction to Trigonometric Identities and Equations
• 7.1 Solving Trigonometric Equations with Identities
• 7.2 Sum and Difference Identities
• 7.3 Double-Angle, Half-Angle, and Reduction Formulas
• 7.4 Sum-to-Product and Product-to-Sum Formulas
• 7.5 Solving Trigonometric Equations
• 7.6 Modeling with Trigonometric Functions
• Introduction to Further Applications of Trigonometry
• 8.1 Non-right Triangles: Law of Sines
• 8.2 Non-right Triangles: Law of Cosines
• 8.3 Polar Coordinates
• 8.4 Polar Coordinates: Graphs
• 8.5 Polar Form of Complex Numbers
• 8.6 Parametric Equations
• 8.7 Parametric Equations: Graphs
• 8.8 Vectors
• Introduction to Systems of Equations and Inequalities
• 9.1 Systems of Linear Equations: Two Variables
• 9.2 Systems of Linear Equations: Three Variables
• 9.3 Systems of Nonlinear Equations and Inequalities: Two Variables
• 9.4 Partial Fractions
• 9.5 Matrices and Matrix Operations
• 9.6 Solving Systems with Gaussian Elimination
• 9.7 Solving Systems with Inverses
• 9.8 Solving Systems with Cramer's Rule
• Introduction to Analytic Geometry
• 10.1 The Ellipse
• 10.2 The Hyperbola
• 10.3 The Parabola
• 10.4 Rotation of Axes
• 10.5 Conic Sections in Polar Coordinates
• Introduction to Sequences, Probability and Counting Theory
• 11.1 Sequences and Their Notations
• 11.2 Arithmetic Sequences
• 11.3 Geometric Sequences
• 11.4 Series and Their Notations
• 11.5 Counting Principles
• 11.6 Binomial Theorem
• 11.7 Probability
• Introduction to Calculus
• 12.1 Finding Limits: Numerical and Graphical Approaches
• 12.2 Finding Limits: Properties of Limits
• 12.3 Continuity
• 12.4 Derivatives
• A | Basic Functions and Identities

3 π 2 3 π 2

7 π 10 7 π 10

α = 150 ° α = 150 °

β = 60 ° β = 60 °

7 π 6 7 π 6

215 π 18 = 37.525  units 215 π 18 = 37.525  units

− 3 π 2 − 3 π 2 rad/s

1655 kilometers per hour

cos ( t ) = − 2 2 , sin ( t ) = 2 2 cos ( t ) = − 2 2 , sin ( t ) = 2 2

cos ( π ) = − 1 , cos ( π ) = − 1 , sin ( π ) = 0 sin ( π ) = 0

sin ( t ) = − 7 25 sin ( t ) = − 7 25

approximately 0.866025403

• ⓐ cos ( 315 ° ) = 2 2 , sin ( 315 ° ) = – 2 2 cos ( 315 ° ) = 2 2 , sin ( 315 ° ) = – 2 2
• ⓑ cos ( − π 6 ) = 3 2 , sin ( − π 6 ) = − 1 2 cos ( − π 6 ) = 3 2 , sin ( − π 6 ) = − 1 2

( 1 2 , − 3 2 ) ( 1 2 , − 3 2 )

sin t = − 2 2 sin t = − 2 2 , cos t = 2 2 cos t = 2 2 , tan t = − 1 tan t = − 1 , sec t = 2 sec t = 2 , csc t = − 2 csc t = − 2 , cot t = − 1 cot t = − 1

sin π 3 = 3 2 sin π 3 = 3 2 , cos π 3 = 1 2 cos π 3 = 1 2 , tan π 3 = 3 tan π 3 = 3 , sec π 3 = 2 sec π 3 = 2 , csc π 3 = 2 3 3 csc π 3 = 2 3 3 , cot π 3 = 3 3 cot π 3 = 3 3

sin ( − 7 π 4 ) = 2 2 , cos ( − 7 π 4 ) = 2 2 , tan ( − 7 π 4 ) = 1 , sin ( − 7 π 4 ) = 2 2 , cos ( − 7 π 4 ) = 2 2 , tan ( − 7 π 4 ) = 1 , sec ( − 7 π 4 ) = 2 , csc ( − 7 π 4 ) = 2 , cot ( − 7 π 4 ) = 1 sec ( − 7 π 4 ) = 2 , csc ( − 7 π 4 ) = 2 , cot ( − 7 π 4 ) = 1

sin t sin t

cos t = − 8 17 , sin t = 15 17 , tan t = − 15 8 cos t = − 8 17 , sin t = 15 17 , tan t = − 15 8 csc t = 17 15 , cot t = − 8 15 csc t = 17 15 , cot t = − 8 15

sin t = − 1 , cos t = 0 , tan t = Undefined sec t =  Undefined, csc t = − 1 , cot t = 0 sin t = − 1 , cos t = 0 , tan t = Undefined sec t =  Undefined, csc t = − 1 , cot t = 0

sec t = 2 , csc t = 2 , tan t = 1 , cot t = 1 sec t = 2 , csc t = 2 , tan t = 1 , cot t = 1

≈ − 2.414 ≈ − 2.414

s i n t = 33 65 , cos t = 56 65 , t a n t = 33 56 , sec t = 65 56 , csc t = 65 33 , cot t = 56 33 s i n t = 33 65 , cos t = 56 65 , t a n t = 33 56 , sec t = 65 56 , csc t = 65 33 , cot t = 56 33

sin ( π 4 ) = 1 2 , cos ( π 4 ) = 1 2 , tan ( π 4 ) = 1 , sin ( π 4 ) = 1 2 , cos ( π 4 ) = 1 2 , tan ( π 4 ) = 1 , sec ( π 4 ) = 2 , c s c ( π 4 ) = 2 , cot ( π 4 ) = 1 sec ( π 4 ) = 2 , c s c ( π 4 ) = 2 , cot ( π 4 ) = 1

adjacent = 10 ; adjacent = 10 ; opposite = 10 3 opposite = 10 3 ; missing angle is π 6 π 6

## 5.1 Section Exercises

Whether the angle is positive or negative determines the direction. A positive angle is drawn in the counterclockwise direction, and a negative angle is drawn in the clockwise direction.

Linear speed is a measurement found by calculating distance of an arc compared to time. Angular speed is a measurement found by calculating the angle of an arc compared to time.

4 π 3 4 π 3

2 π 3 2 π 3

7 π 2 ≈ 11.00  in 2 7 π 2 ≈ 11.00  in 2

81 π 20 ≈ 12.72  cm 2 81 π 20 ≈ 12.72  cm 2

− 3 π − 3 π radians

5 π 6 5 π 6 radians

5.02 π 3 ≈ 5.26 5.02 π 3 ≈ 5.26 miles

25 π 9 ≈ 8.73 25 π 9 ≈ 8.73 centimeters

21 π 10 ≈ 6.60 21 π 10 ≈ 6.60 meters

104.7198 cm 2

0.7697 in 2

8 π 9 8 π 9

7 in./s, 4.77 RPM, 28.65 deg/s

1 , 809 , 557.37  mm/min = 30.16  m/s 1 , 809 , 557.37  mm/min = 30.16  m/s

5.76 5.76 miles

120 ° 120 °

794 miles per hour

2,234 miles per hour

11.5 inches

## 5.2 Section Exercises

The unit circle is a circle of radius 1 centered at the origin.

Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle, t , t , formed by the terminal side of the angle t t and the horizontal axis.

The sine values are equal.

60° , 60° , Quadrant IV, sin ( 300° ) = − 3 2 , cos ( 300° ) = 1 2 sin ( 300° ) = − 3 2 , cos ( 300° ) = 1 2

45° , 45° , Quadrant II, sin ( 135° ) = 2 2 , sin ( 135° ) = 2 2 , cos ( 135° ) = − 2 2 cos ( 135° ) = − 2 2

60° , 60° , Quadrant II, sin ( 120° ) = 3 2 , sin ( 120° ) = 3 2 , cos ( 120° ) = − 1 2 cos ( 120° ) = − 1 2

30° , 30° , Quadrant II, sin ( 150° ) = 1 2 , sin ( 150° ) = 1 2 , cos ( 150° ) = − 3 2 cos ( 150° ) = − 3 2

π 6 , π 6 , Quadrant III, sin ( 7 π 6 ) = − 1 2 , sin ( 7 π 6 ) = − 1 2 , cos ( 7 π 6 ) = − 3 2 cos ( 7 π 6 ) = − 3 2

π 4 , π 4 , Quadrant II, sin ( 3 π 4 ) = 2 2 , sin ( 3 π 4 ) = 2 2 , cos ( 4 π 3 ) = − 2 2 cos ( 4 π 3 ) = − 2 2

π 3 , π 3 , Quadrant II, sin ( 2 π 3 ) = 3 2 , sin ( 2 π 3 ) = 3 2 , cos ( 2 π 3 ) = − 1 2 cos ( 2 π 3 ) = − 1 2

π 4 , π 4 , Quadrant IV, sin ( 7 π 4 ) = − 2 2 , sin ( 7 π 4 ) = − 2 2 , cos ( 7 π 4 ) = 2 2 cos ( 7 π 4 ) = 2 2

− 15 4 − 15 4

( − 10 , 10 3 ) ( − 10 , 10 3 )

( – 2.778 , 15.757 ) ( – 2.778 , 15.757 )

[ – 1 , 1 ] [ – 1 , 1 ]

sin t = 1 2 , cos t = − 3 2 sin t = 1 2 , cos t = − 3 2

sin t = − 2 2 , cos t = − 2 2 sin t = − 2 2 , cos t = − 2 2

sin t = 3 2 , cos t = − 1 2 sin t = 3 2 , cos t = − 1 2

sin t = − 2 2 , cos t = 2 2 sin t = − 2 2 , cos t = 2 2

sin t = 0 , cos t = − 1 sin t = 0 , cos t = − 1

sin t = − 0.596 , cos t = 0.803 sin t = − 0.596 , cos t = 0.803

sin t = 1 2 , cos t = 3 2 sin t = 1 2 , cos t = 3 2

sin t = − 1 2 , cos t = 3 2 sin t = − 1 2 , cos t = 3 2

sin t = 0.761 , cos t = − 0.649 sin t = 0.761 , cos t = − 0.649

sin t = 1 , cos t = 0 sin t = 1 , cos t = 0

− 6 4 − 6 4

( 0 , – 1 ) ( 0 , – 1 )

37.5 seconds, 97.5 seconds, 157.5 seconds, 217.5 seconds, 277.5 seconds, 337.5 seconds

## 5.3 Section Exercises

Yes, when the reference angle is π 4 π 4 and the terminal side of the angle is in quadrants I and III. Thus, at x = π 4 , 5 π 4 , x = π 4 , 5 π 4 , the sine and cosine values are equal.

Substitute the sine of the angle in for y y in the Pythagorean Theorem x 2 + y 2 = 1. x 2 + y 2 = 1. Solve for x x and take the negative solution.

The outputs of tangent and cotangent will repeat every π π units.

2 3 3 2 3 3

− 2 3 3 − 2 3 3

− 3 3 − 3 3

If sin t = − 2 2 3 sin t = − 2 2 3 , sec t = − 3 sec t = − 3 , csc t = − 3 2 4 csc t = − 3 2 4 , tan t = 2 2 tan t = 2 2 , cot t = 2 4 cot t = 2 4

sec t = 2 sec t = 2 , csc t = 2 3 3 csc t = 2 3 3 , tan t = 3 tan t = 3 , cot t = 3 3 cot t = 3 3

− 2 2 − 2 2

sin t = 2 2 , cos t = 2 2 , tan t = 1 , cot t = 1 , sec t = 2 , csc t = 2 sin t = 2 2 , cos t = 2 2 , tan t = 1 , cot t = 1 , sec t = 2 , csc t = 2

sin t = − 3 2 sin t = − 3 2 , cos t = − 1 2 cos t = − 1 2 , tan t = 3 , cot t = 3 3 tan t = 3 , cot t = 3 3 , sec t = − 2 sec t = − 2 , csc t = − 2 3 3 csc t = − 2 3 3

sin ( t ) ≈ 0.79 sin ( t ) ≈ 0.79

csc t ≈ 1.16 csc t ≈ 1.16

sin t cos t = tan t sin t cos t = tan t

13.77 hours, period: 1000 π 1000 π

7.73 inches

## 5.4 Section Exercises

The tangent of an angle is the ratio of the opposite side to the adjacent side.

For example, the sine of an angle is equal to the cosine of its complement; the cosine of an angle is equal to the sine of its complement.

b = 20 3 3 , c = 40 3 3 b = 20 3 3 , c = 40 3 3

a = 10 , 000 , c = 10 , 000.5 a = 10 , 000 , c = 10 , 000.5

b = 5 3 3 , c = 10 3 3 b = 5 3 3 , c = 10 3 3

5 29 29 5 29 29

5 41 41 5 41 41

c = 14 , b = 7 3 c = 14 , b = 7 3

a = 15 , b = 15 a = 15 , b = 15

b = 9.9970 , c = 12.2041 b = 9.9970 , c = 12.2041

a = 2.0838 , b = 11.8177 a = 2.0838 , b = 11.8177

a = 55.9808 , c = 57.9555 a = 55.9808 , c = 57.9555

a = 46.6790 , b = 17.9184 a = 46.6790 , b = 17.9184

a = 16.4662 , c = 16.8341 a = 16.4662 , c = 16.8341

498.3471 ft

368.7633 ft

− 7 π 6 − 7 π 6

10.385 meters

2 π 11 2 π 11

1036.73 miles per hour

2 2 2 2 or − 2 2 − 2 2

sine, cosecant, tangent, cotangent

b = 8 , c = 10 b = 8 , c = 10

11 157 157 11 157 157

a = 4 , b = 4 a = 4 , b = 4

6.283 centimeters

3.351 feet per second, 2 π 75 2 π 75 radians per second

− 3 2 − 3 2

a = 9 2 , b = 9 3 2 a = 9 2 , b = 9 3 2

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## Tangent And Cotangent

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## Tangent And Cotangent

Displaying top 8 worksheets found for - Tangent And Cotangent .

Some of the worksheets for this concept are 11 secant tangent and tangent tangent angles, Tangent ratio classwork work, Tangent lines date period, Work the tangent line problem, Graphing tangent transformations, Cosecant secant cotangent, Sine cosine and tangent practice, 2 4 1 precalculus work name 1 tangent and.

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## 1. 11-Secant-Tangent and Tangent-Tangent Angles

2. tangent ratio classwork worksheet, 3. tangent lines date period, 4. worksheet { the tangent line problem, 5. graphing tangent transformations -, 6. cosecant, secant & cotangent, 7. sine, cosine, and tangent practice, 8. 2 4 1 precalculus worksheet 4.5 name 1 =- tangent and ....

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1. Precalculus

Lesson 6.5 Graphs of Tangent and Cotangent Functions Annotated Notes 6.5 Part 2 Worksheet 6.5 Part 2 Graphing Tangent and Cotangent Video Lesson 6.5 this covers both part 1 and part 2: Secant, Cosecant, Tangent and Cotangent Functions Lesson 6.6 Phase Shift Annotated Notes 6.6 Video Lesson 6.6 Worksheet 6.6 Lesson 8.5 Simple Harmonic Motion

2. PDF Pre-Calculus Worksheet Name: Tangent and Cotangent Per:

1 2 3. = − 5tan ( 3 x + π 2. y = ) Amplitude: Amplitude: Amplitude: Flip? Vertical Shift: Period: Phase Shift: Flip? Vertical Shift: Period: Phase Shift: Flip? Vertical Shift: Period: Phase Shift: II. Graph each function, over one period, showing the vertical asymptotes. 10. y = − 1 cot x − 1 3 11. y = 2 tan

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Precalculus: Graphs of Tangent, Cotangent, Secant, and Cosecant Practice Problems Questions 1. Describe the graph of the function in terms of basic trigonometric functions. Locate the vertical asymptotes and sketch two periods of the function. y = 3tan(x/2) 2. Solve the equation cscx = 1 in the interval 2π ≤ x ≤ 5π/2.

4. Precalculus

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OpenStax OpenStax Learning Objectives Analyze the graph of y = tanx. Graph variations of y = tanx. Analyze the graphs of y = secx and y = cscx. Graph variations of y = secx and y = cscx. Analyze the graph of y = cotx. Graph variations of y = cotx.

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Precalculus Worksheet and Notes Covering• Graphing Tangent and Cotangent• Midline• Amplitude• Period• Phase ShiftsYou will receive a worksheet as well as fill in the blank notes with the purchase of this resource. Students will practice the necessary skills of graphing trig equations to be successf...

10. Precalculus & Trigonometry

Precalculus 1e (OpenStax) Precalculus - An Investigation of Functions (Lippman and Rasmussen) APEX PreCalculus (Chapman, Herald, and Libertini) Precalculus (Tradler and Carley) Active Prelude to Calculus (Boelkins) Corequisite Companion to Precalculus (Freidenreich) Precalculus (Stitz-Zeager) Elementary Trigonometry (Corral) Trigonometry ...

11. The Other Trigonometric Functions

The secant, cotangent, and cosecant are all reciprocals of other functions. The secant is the reciprocal of the cosine function, the cotangent is the reciprocal of the tangent function, and the cosecant is the reciprocal of the sine function. The six trigonometric functions can be found from a point on the unit circle.

12. Precalculus Worksheets

Study.com's precalculus worksheets cover topics like continuity, limits, rate of change, vectors, matrices, and more. ... Quiz & Worksheet - Cotangent. 21K. ... Quiz & Worksheet - Tangent & the ...

1. cos(t) = − 2√ 2, sin(t) = 2√ 2 cos ( t) = − 2 2, sin ( t) = 2 2 2. cos(π) = −1, cos ( π) = − 1, sin(π) = 0 sin ( π) = 0 3. sin(t) = − 725 sin ( t) = − 7 25 4. approximately 0.866025403 5. π 3 π 3 6.

14. Precalculus: Graphs of the Tangent, Cotangent, Cosecant, and Secant

Four examples of how to graph the tangent, Cotangent, secant, and cosecant functions involving vertical stretching, reflecting, shifting, and horizontal stre...

15. Precalculus Examples

Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor.

16. PRE 10

1. 1 2 tan 3 2 y x = 2 . 4 cot 4 1 3 2 y x π = − 3 . ( ) 5 tan 3 y x π = + Amplitude : Flip ? Vertical Shift : Period : Phase Shift : Amplitude : Flip ? Vertical Shift : Period : Phase Shift : Amplitude : Flip ? Vertical Shift : Period : Phase Shift : II.

17. Tangent And Cotangent Teaching Resources

Graphing the tangent and cotangent functions can be difficult for students. This 26 page animated interactive PowerPoint leads your students through the process of graphing tangent and cotangent functions step by step, including determining Amplitude, Period, Phase Shift, and Asymptotes.Also included in this resource is a Handout which includes graphing and questions on amplitude, period ...

18. Tangent And Cotangent Worksheets

Some of the worksheets for this concept are 11 secant tangent and tangent tangent angles, Tangent ratio classwork work, Tangent lines date period, Work the tangent line problem, Graphing tangent transformations, Cosecant secant cotangent, Sine cosine and tangent practice, 2 4 1 precalculus work name 1 tangent and.

19. Tangent Graphing.pdf

View Tangent Graphing.pdf from MATH N/A at IPreparatory Academy - Florida. Pre-Calculus Worksheet Tangent and Cotangent Name: _ Per: _ I. Fill in the chart for each function. DO NOT GRAPH. Remember

20. Results for graphing tangent and cotangent

Precalculus Worksheet and Notes Covering• Graphing Tangent and Cotangent • Midline• Amplitude• Period• Phase ShiftsYou will receive a worksheet as well as fill in the blank notes with the purchase of this ... These notes cover the graphs of tangent, cotangent, secant and cosecant and their properties (domain, range, period, vertical ...

21. Tangent and Cotangent Graphs Worksheet.doc

Precalculus/Trig 3 Name _____ Tangent & Cotangent Graphs Worksheet Without using a calculator. Identify the amplitude, period, all VAs, and any shifts or reflections that occur on the graph. Sketch 2 periods in radians.

22. Tangent And Cotangent Worksheets

Displaying top 8 worksheets found for - Tangent And Cotangent. Some of the worksheets for this concept are 11 secant tangent and tangent tangent angles, Tangent ratio classwork work, Tangent lines date period, Work the tangent line problem, Graphing tangent transformations, Cosecant secant cotangent, Sine cosine and tangent practice, 2 4 1 precalculus work name 1 tangent and.