6.1 Reciprocal, Quotient, and Pythagorean Identities.notebook. Chapter 6: Trigonometric Identities

Transcription

1 Chapter 6: Trigonometric Identities 1

2 Chapter 6 Complete the following table: 6.1 Reciprocal, Quotient, and Pythagorean Identities Pages Proving Identities Pages Measure of <A 0 o 30 o 45 o 60 o 90 o sin A cos A sin 2 A + cos 2 A What conclusion do you make? 2

3 Trigonometric Identity a trigonometric equation that is true for all permissible values of the variable expressions on both sides of the equation. Pythagorean Identities: Reciprocal Identities: Quotient Identities: 3

4 NOTE: All identities can be rearranged to suit the problem at hand. 4

5 Example 1 Page 291 Verify a Potential Identity Numerically and Graphically a) Determine the non permissible values, in degrees, for the equation. b) Numerically verify that θ = 60 and θ = are solutions of the equation. c) Use technology to graphically decide whether the equation could be an identity over the domain 360 < θ

6 Example 1: Your Turn Page 293 a) Determine the non permissible values, in degrees, for the equation b) Verify that x = 45 and x = are solutions to the equation. c) Use technology to graphically decide whether the equation could be an identity over the domain 360 < x 360. Answer a b F c 6

7 Example 2 Page 293 Use Identities to Simplify Expressions a) Determine the non permissible values, in radians, of the variable in b) Simplify the expression. 7

8 Example 2: Your Turn Page 294 a) Determine the non permissible values, in radians, of the variable in the expression b) Simplify the expression. Answer 8

9 Example 3 Page 295 Use the Pythagorean Identity a) Verify that the equation cot 2 x + 1 = csc 2 x is true when x = b) Use quotient identities to prove cot 2 x + 1 = csc 2 x. 9

10 Example 3: Your Turn Page 295 a) Verify the equation 1 + tan 2 x = sec 2 x numerically for x = b) Use quotient identities to prove 1 + tan 2 x = sec 2 x. Answer 10

11 More Examples: 1. Simplify the following: a) b) 11

12 2. Using : a) Verify it is true for b) Prove the identity. c) State and non permissible value. 12

13 Example 1 page 310 Verify Versus Prove That an Equation Is an Identity a) Verify that 1 sin 2 x = sin x cos x cot x for some values of x. Determine the non permissible values for x. Work in degrees. b) Prove that 1 sin 2 x = sin x cos x cot x for all permissible values of x. 13

14 Example 1: Your Turn Page 311 a) Determine the non permissible values for the equation b) Verify that the equation may be an identity, either graphically using technology or by choosing one value for x. c) Prove that the identity is true for all permissible values of x. Answer 14

15 Example 3 Page 312 Prove that values of x. is an identity for all permissible 15

16 Some helpful hints when simplifying or proving identities: > when working with identities, keep left side and right side separate NEVER cross the equals sign and work with more complicated side of the equation > if there are squared terms, check to see if a Pythagorean identity can be applied. > express in terms of sine and cosine > if above does not work, multiply by an expression equivalent to 1 (same idea as rationalizing) 16