Chapter5: Trigonometric Equations and Identities
Chapter5 Readiness Check
Prerequisites
Students should be able to:
· use a graphing calculator to solve an equation using either the intersect feature or the zero
feature, and identify the period of a trigonometric function from its equation. [5.1]
· work with fraction involving p and visualize how to divide a specified length into halves, thirds or quarters. [5.2]
· work with unit circle definitions of the trigonometric functions and reciprocal functions. [5.3]
· verify trigonometric identities numerically and graphically and prove them algebraically. [5.4]
· use the sum and difference identities for sine and cosine, simplify trigonometric expressions, and
prove identities. [5.5]
· use the order of operations involving trigonometric functions. [5.6]
Readiness Check Exercises for Chapter5
1. Define the following terms: cycle, exact value, periodic function, quarter-cycle, roots of an equation, trigonometric function.
2. Solve x2=x+2 by:
a) graphing y=x2 and y=x+2
b) graphing y=x2-x-2
c) using algebra.
3. Determine the period of each function.
a) y=sin2x b) y=cos0.5x c) y=tan3x
4. Divide each length into thirds. Use a common denominator to write the value of the midpoint and the value of the endpoints.
a) ┌──────────┐ b) ┌──────────┐
0 0
5. Divide each length into thirds. Use a common denominator to write the value of the midpoint and the value of the endpoints.
a) ┌──────────┐ b) ┌──────────┐
0 0
6. Simplify.
a) +p b) + c) +
7. Write the definition of each reciprocal trigonometric function.
a) cscx b) secx c) cotx
8. Simplify.
a) a´´ b) + c)
Chapter5 Overview
There are nine main objectives to be covered in this chapter:
1. A trigonometric equation can be solved graphically and algebraically.
2. The particular solution of a trigonometric equation can be found for any specified
domain.
3. The general solution of a trigonometric equation can be found where the domain is the set
of real numbers.
4. To determine exact solutions of a trigonometric equation, it must be solved algebraically.
5. A trigonometric identity is a trigonometric equation that is satisfied for all values of the
variable for which both sides of the equation are defined.
6. A trigonometric identity can be verified numerically and graphically.
7. A trigonometric identity can be proven algebraically.
8. To prove a trigonometric identity, it is necessary to use basic identities and algebraically
manipulate one side of the identity until is it equivalent to the other side. It is also correct
to show that both sides of the identity simplify to the same expression. However, it is
essential that each side is simplified independently of the other side.
9. The sum, difference and double-angle identities for sine and cosine can be used to verify
trigonometric identities and simplify trigonometric expressions.
Chapter5 Assignments
5.1 Solving Trigonometric Equations Using Graphing Technology
In this section, you will solve trigonometric equations using graphing technology.
Read pages298301, making notes on important ideas.
Do exercises #1, 5, 811, 16 on pages302304.
5.2 Solving Trigonometric Equations without Using Graphing TechnologIn this section, you will solve trigonometric equations that have exact solutions.
Read pages308312, making notes on important ideas.
Do exercises #16, 9, 11 on pages313314.
5.3 Trigonometric Identities
In this section, you will recognize and verify specific trigonometric identities numerically and graphically.
Read pages315318, making notes on important ideas.
Do exercises #110, 12 on pages319321.
5.4 Verifying and Proving Trigonometric Identities
In this section, you will verify trigonometric identities numerically and graphically, and then prove them algebraically.
Read pages322325, making notes on important ideas.
Do exercises #39, 1213 on pages326327.
5.5 Sum and Difference Identities
In this section, you will use the sum and difference identities for sine and cosine to verify
And simplify trigonometric expressions, and to prove identities.
Read pages329333, making notes on important ideas.
Do. exercises #36, 10, 1215, 1718 on pages333335.
5.6 Identities for sin2x and cos2x
In this section, you will use the identities for sin2x and cos2x.
Read pages338341, making notes on important ideas.
Do exercises #3, 58, 1014, 17, 19, 24 on pages342345.
Chapter5Review
Reread the main objectives listed at the beginning of this section.
Make sure you know the following key terms:
· double-angle identities
· exact from of the solution
· general solution of an equation
· identity
· odd-even identities
· Pythagorean identity
· quotient identity
· sum and difference identities
· trigonometric equation
· trigonometric
· identity
Try to do all the questions in the Chapter Review without looking at your notes or text. If you need to consult your notes for certain questions, it shows you may need extra work in those areas. Go back to the relevant section to clarify and review. If you need more help or extra questions, ask your teacher.
Complete the Chapter5 Review, #112 on page 349, write all your steps. Ask your teacher for the test when you are ready.