Analytic Trigonometry

Transcription

1 Chapter 5 Analytic Trigonometry Course Number 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 (Page 35) Name four ways in which the fundamental trigonometric identities can be used: 1) to evaluate trigonometric functions ) to simplify trigonometric expressions 3) to develop additional trigonometric identities 4) to solve trigonometric equations Instructor Date How to recognize and write the fundamental trigonometric identities The Fundamental Trigonometric Identities List six reciprocal identities: 1) sin u = 1/(csc u) ) cos u = 1/(sec u) 3) tan u = 1/(cot u) 4) csc u = 1/(sin u) 5) sec u = 1/(cos u) 6) cot u = 1/(tan u) List two quotient identities: 1) tan u = (sin u)/(cos u) ) cot u = (cos u)/(sin u) List three Pythagorean identities: 1) sin u + cos u = 1 ) 1 + tan u = sec u 3) 1 + cot u = csc u List six cofunction identities: 1) sin(π/ u) = cos u ) cos(π/ u) = sin u 3) tan(π/ u) = cot u 4) cot(π/ u) = tan u 5) sec(π/ u) = csc u 6) csc(π/ u) = sec u List six even/odd identities: 1) sin( u) = sin u ) cos( u) = cos u 3) tan( u) = tan u 4) csc( u) = csc u 5) sec( u) = sec u 6) cot( u) = cot u 83

2 84 Chapter 5 Analytic Trigonometry II. Using the Fundamental Identities (Pages ) Example 1: Explain how to use the fundamental trigonometric identities to find the value of tan u given that sec u =. Use the Pythagorean identity 1 + tan u = sec u. Substitute for the value of sec u and solve for tan u. How to use the fundamental trigonometric identities to evaluate trigonometric functions, simplify trigonometric expressions, and rewrite trigonometric expressions Example : Explain how to use the fundamental trigonometric identities to simplify sec x tan x sin x. Rewrite the expression in terms of sines and cosines. Combine the resulting fractions to obtain (1 sin x)/(cos x). Using the Pythagorean identity sin u + cos u = 1, replace the numerator with cos x. Simplify the result to obtain cos x. Example 3: Explain how to use a graphing utility to verify 3 whether sec x sin x + sin x cos x = tan x is an identity. Graph y 1 = sec x sin 3 x + sin x cos x and y = tan x in the same viewing window. If the two graphs appear to coincide, the expressions appear to be equivalent and the equation is an identity. If the two graphs do not coincide, then the equation is not an identity. Homework Assignment Page(s) Exercises

3 Section 5. Verifying Trigonometric Identities 85 Section 5. Verifying Trigonometric Identities Objective: In this lesson you learned how to verify trigonometric identities. Course Number Instructor Date I. Introduction (Page 360) The key to both verifying identities and solving equations is... the ability to use the fundamental identities and the rules of algebra to rewrite trigonometric expressions. How to understand the difference between conditional equations and identities An identity is... an equation that is true for all real values in the domain of the variable. II. Verifying Trigonometric Identities (Pages ) Complete the following list of guidelines for verifying trigonometric identities: How to verify trigonometric identities 1) Work with one side of the equation at a time. It is often better to work with the more complicated side first. ) Look for opportunities to factor an expression, add fractions, square a binomial, or create a monomial denominator. 3) Look for opportunities to use the fundamental identities. Note which functions are in the final expression you want. Sines and cosines pair up well, as do secants and tangents, and cosecants and cotangents. 4) If the preceding guidelines do not help, try converting all terms to sines and cosines. 5) Always try something! Even making an attempt that leads to a dead end provides insight. Example 1: Describe a strategy for verifying the identity sin θ tanθ + cosθ = secθ. Then verify the identity. Begin by converting all terms to sines and cosines.

4 86 Chapter 5 Analytic Trigonometry Example : Describe a strategy for verifying the identity sin x(csc x 1)(csc x + 1) = 1 sin verify the identity. x. Then Because the left side is more complicated, start with it. Begin by multiplying (csc x 1) by (csc x + 1), and then search for a fundamental identity that can be used to replace the result. Example 3: Verify the identity 5 3 cot α = cot α csc α cot α. 3 Additional notes Homework Assignment Page(s) Exercises

5 Section 5.3 Solving Trigonometric Equations 87 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. Course Number Instructor Date I. Introduction (Pages ) To solve a trigonometric equation,... use standard algebraic techniques such as collecting like terms and factoring. How to use standard algebraic techniques to solve trigonometric equations The preliminary goal in solving trigonometric equations is... to isolate the trigonometric function involved in the equation. How many solutions does the equation sec x = have? Explain. The equation has an infinite number of solutions because the secant function has a period of π. Any angles coterminal with the equation s solutions on [0, π) will also be solutions of the equation. Example 1: Solve cos x 1 = 0. x = π/4 + nπ, x = 3π/4 + nπ To solve an equation in which two or more trigonometric functions occur,... collect all terms on one side and try to separate the functions by factoring or by using appropriate identities. II. Equations of Quadratic Type (Pages ) Give an example of a trigonometric equation of quadratic type. Answers will vary. For example, cos x + 4 cos x + 4 = 0. How to solve trigonometric equations of quadratic type To solve a trigonometric equation of quadratic type,... factor the quadratic, or if factoring is not possible, use the Quadratic Formula.

6 88 Chapter 5 Analytic Trigonometry Example : Solve tan x + tan x = 1. x = 3π/4 + nπ Care must be taken when squaring each side of a trigonometric equation to obtain a quadratic because... this procedure can introduce extraneous solutions, so any solutions must be checked in the original equation to see whether they are valid or extraneous. III. Functions Involving Multiple Angles (Page 373) Give an example of a trigonometric function of multiple angles. Answers will vary. For example, tan 4x. How to solve trigonometric equations involving multiple angles Example 3: Solve sin 4x =. x = π/16 + nπ/ and x = 3π/16 + nπ/ IV. Using Inverse Functions (Page ) Example 4: Use inverse functions to solve the equation tan x + 4 tan x + 4 = 0. x = arctan ( ) + nπ How to use inverse trigonometric functions to solve trigonometric equations Homework Assignment Page(s) Exercises

7 Section 5.4 Sum and Difference Formulas 89 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. Course Number Instructor Date I. Using Sum and Difference Formulas (Pages ) List the sum and difference formulas for sine, cosine, and tangent. sin(u + v) = sin u cos v + cos u sin v sin(u v) = sin u cos v cos u sin v cos(u + v) = cos u cos v sin u sin v cos(u v) = cos u cos v + sin u sin v tan(u + v) = (tan u + tan v)/(1 tan u tan v) tan(u v) = (tan u tan v)/(1 + tan u tan v) How to use sum and difference formulas to evaluate trigonometric functions, to verify identities, and to solve trigonometric equations Example 1: Use a sum or difference formula to find the exact value of tan 55. ( )/6 Example : Find the exact value of cos 95 cos 35 + sin 95 sin 35. 1/ A reduction formula is... a formula involving expressions such as sin(θ + nπ/) or cos(θ + nπ/), where n is an integer, that can be derived from sum and difference formulas. Example 3: Derive a reduction formula for π sin t +. sin(t + π/) = cos t

8 90 Chapter 5 Analytic Trigonometry π π Example 4: Find all solutions of cos( x ) + cos( x + ) = in the interval [0, π). x = 0 Additional notes Homework Assignment Page(s) Exercises

9 Section 5.5 Multiple-Angle and Product-to-Sum Formulas 91 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. Course Number Instructor Date I. Multiple-Angle Formulas (Pages ) The most commonly used multiple-angle formulas are the double-angle formulas, which are listed below: sin u = sin u cos u How to use multipleangle formulas to rewrite and evaluate trigonometric functions cos u = cos u sin u = cos u 1 = 1 sin u tan u = ( tan u)/(1 tan u) To obtain other multiple-angle formulas,... use 4θ and θ or 6θ and 3θ in place of θ and θ in the double-angle formulas or using the double-angle formulas together with the appropriate trigonometric sum formulas. Example 1: Use multiple-angle formulas to express cos 3x in terms of cos x. 4 cos 3 x 3 cos x II. Power-Reducing Formulas (Page 389) The double-angle formulas can be used to obtain the power-reducing formulas. How to use powerreducing formulas to rewrite and evaluate trigonometric functions

10 9 Chapter 5 Analytic Trigonometry The power-reducing formulas are: sin u = cos u = (1 cos u)/ (1 + cos u)/ tan u = (1 cos u)/(1 + cos u) III. Half-Angle Formulas (Pages ) List the half-angle formulas: sin u = ± (1 cos u)/ How to use half-angle formulas to rewrite and evaluate trigonometric functions cos u = ± (1 + cos u)/ tan u = (1 cos u)/(sin u) = (sin u)/(1 + cos u) The signs of sin (u/) and cos (u/) depend on... quadrant in which u/ lies. the Example : Find the exact value of tan IV. Product-to-Sum Formulas (Pages ) The product-to-sum formulas are used in calculus to... evaluate integrals involving the products of sines and cosines of two different angles. How to use product-tosum and sum-to-product formulas to rewrite and evaluate trigonometric functions The product-to-sum formulas are: sin u sin v = 1/[cos(u v) cos (u + v)] cos u cos v= sin u cos v = cos u sin v = 1/[cos(u v) + cos (u + v)] 1/[sin(u + v) + sin(u v)] 1/[sin(u + v) sin(u v)]

11 Section 5.5 Multiple-Angle and Product-to-Sum Formulas 93 Example 3: Write cos 3x cos x as a sum or difference. 1/ cos x + 1/ cos 5x The sum-to-product formulas can be used to... rewrite a sum or difference of trigonometric functions as a product. The sum-to-product formulas are: sin u + sin v = sin u sin v = cos u + cos v = cos u cos v = sin((u + v)/) cos((u v)/) cos((u + v)/) sin((u v)/) cos((u + v)/) cos((u v)/) sin((u + v)/) sin((u y)/) Example 4: Write cos 4x + cos x as a sum or difference. cos 3x cos x Additional notes

12 94 Chapter 5 Analytic Trigonometry Additional notes Homework Assignment Page(s) Exercises