Analytic Trigonometry

Transcription

1 Name Chapter 5 Analytic Trigonometry Section 5.1 Using Fundamental Identities Objective: In this lesson you learned how to use fundamental trigonometric identities to evaluate trigonometric functions and simplify trigonometric expressions. I. Introduction Name four ways in which the fundamental trigonometric identities can be used: 1) How to recognize and write the fundamental trigonometric identities 2) 3) 4) List the Fundamental Trigonometric Identities List the six reciprocal identities List the six co-function identities 1) 1) 2) 2) 3) 3) 4) 4) 5) 5) 6) 6) 1

2 List the two quotient identities List the six even/odd identities 1) 1) 2) 2) List the three Pythagorean identities 3) 1) 2) 4) 3) 5) 6) II. Using the Fundamental Identities Example 1: Explain how to use the fundamental trigonometric identities to find the value of tan u given that sec u = 2. How to use the fundamental trigonometric identities to evaluate trigonometric functions, simplify trigonometric expressions, and rewrite trigonometric expressions Example 2: Explain how to use the fundamental trigonometric identities to simplify sec x tan x sin x. 2

3 Section 5.1 Examples Using Fundamental Identities ( 1 ) Use the given values to evaluate (if possible) all six trigonometric functions. sin x = 1 2 cos x = 3 2 ( 2 ) Use the fundamental identities to simplify the expression. sin θ (csc θ sin θ) ( 3 ) Factor the expression and use the fundamental identities to simplify. cot 2 x cot 2 x cos 2 x ( 4 ) Perform the multiplication and use the fundamental identities to simplify. (sin x + cos x) 2 ( 5 ) Use trigonometric substitution to write the algebraic expression as a trigonometric function of θ, where 0 < θ < π x 2, x = 5 sin θ 3

4 Section 5.2 Verifying Trigonometric Identities Objective: In this lesson you learned how to verify trigonometric identities I. Introduction The key to both verifying identities and solving equations is: How to understand the difference between conditional equations and identities An identity is: II. Verifying Trigonometric Identities Complete the following list of guidelines for verifying trigonometric identities: 1) How to verify trigonometric identities 2) 3) 4) 5) 4

5 III. Exponent Properties Review Complete the following: a m a n = (a m ) n = a m a n = a n = a 0 = 5

6 Section 5.2 Examples Verifying Trigonometric Identities ( 1 ) Verify the identity. a) sin t csc t = 1 b) sin 1 2 x cos x sin 5 2 x cos x = cos 3 x sin x c) cos θ 1 sin θ = sec θ + tan θ d) 2 sec 2 x 2 sec 2 x sin 2 x sin 2 x cos 2 x = 1 6

7 Section 5.3 Solving Trigonometric Equations Objective: In this lesson you learned how to use standard algebraic techniques and inverse trigonometric functions to solve trigonometric equations. I. Introduction To solve a trigonometric equation: How to use standard algebraic techniques to solve trigonometric equations The preliminary goal in solving trigonometric equations is: How many solutions does the equation sec x = 2 have? Explain. To solve an equation in which two or more trigonometric functions occur: II. Equations of a Quadratic Type Give an example of a trigonometric equation of a quadratic type. How to solve trigonometric equations of quadratic type To solve a trigonometric equation of quadratic type: Care must be taken when squaring each side of a trigonometric equation to obtain a quadratic because: III. Functions Involving Multiple Angles Give an example of a trigonometric function of multiple angles. How to solve trigonometric equations involving multiple angles 7

8 Section 5.3 Examples Solving Trigonometric Equations ( 1 ) Verify that each x-value is a solution of the equation. 2 cos x 1 = 0 a) x = π 3 b) x = 5π 3 ( 2 ) Find all solutions of the equation in the intervals [0, 360 ) and [0, 2π). sin x = 2 2 ( 3 ) Solve the equation. 3 sec 2 x 4 = 0 ( 4 ) Find all solutions of the equation in the interval [0, 2π). cos 3 x = cos x 8

9 Section 5.4 Sum and Difference Formulas Objective: In this lesson you learned how to use sum and difference formulas to rewrite and evaluate trigonometric functions. I. Using Sum and Difference Formulas List the sum and difference formulas for sine, cosine, and tangent. sin(u + v) = sin(u v) = How to use sum and difference formulas to evaluate trigonometric functions, to verify identities and to solve trigonometric equations cos(u + v) = cos(u v) = tan(u + v) = tan(u v) = A reduction formula is: 9

10 Section 5.4 Examples Sum and Difference Formulas ( 1 ) Find the exact value of each expression. a) cos(240 0 ) b) cos 240 cos 0 ( 2 ) Find the exact values of the sine, cosine, and tangent of the angle. 165 = ( 3 ) Write the expression as the sine, cosine, or tangent of an angle. cos 60 cos 10 sin 60 sin 10 ( 4 ) Find the exact value of the expression without using a calculator. sin [ π 2 + sin 1 ( 1)] 10

11 Section 5.5 Multiple-Angle and Product-to-Sum Formulas Objective: In this lesson you learned how to use multiple-angle formulas, power-reducing formulas, half-angle formulas, and product-to-sum formulas to rewrite and evaluate trigonometric functions. I. Multiple-Angle Formulas The most commonly used multiple-angle formulas are the, which are listed below: sin 2u = How to use multiple-angle formulas to rewrite and evaluate trigonometric functions cos 2u = = = tan 2u = To obtain other multiple-angle formulas: II. Power-Reducing Formulas The double-angle formulas can be used to obtain the. The power-reducing formulas are: How to use power-reducing formulas to rewrite and evaluate trigonometric functions sin 2 u = cos 2 u = tan 2 u = 11

12 III. Half-Angle Formulas List the half-angle formulas: sin u 2 = cos u 2 = How to use half-angle formulas to rewrite and evaluate trigonometric functions tan u = = 2 The signs of sin u and cos u depend on: 2 2 IV. Product-to-Sum Formulas The product-to-sum formulas are used in calculus to: The product-to-sum formulas are: How to use product-to-sum and sum-to-product formulas to rewrite and evaluate trigonometric functions sin u sin v = sin u cos v = cos u cos v = cos u sin v = The sum-to-product formulas can be used to: The sum-to-product formulas are: sin u + sin v = cos u + cos v = sin u sin v = cos u cos v = 12

13 Section 5.5 Examples Multiple-Angle and Product-to-Sum Formulas ( 1 ) Find the exact values of sin 2u, cos 2u, and tan 2u using the double-angle formulas. sin u = 3 5, 0 < u < π 2 ( 2 ) Use a double-angle formula to rewrite the expression. 8 sin x cos x ( 3 ) Find the exact values of sin u, cos u, and tan u using the half-angle formulas cos u = 3 5, 0 < u < π 2 ( 4 ) Find all solutions of the equation in the interval [0, 2π). sin 6x + sin 2x = 0 13