Trigonometry LESSON SIX - Trigonometric Identities I Lesson Notes

Transcription

1 LESSON SIX - Trigonometric Identities I Example Understanding Trigonometric Identities. a) Why are trigonometric identities considered to be a special type of trigonometric equation? Trigonometric Identities A trigonometric equation that IS an identity: A trigonometric equation that is NOT an identity: b) Which of the following trigonometric equations are also trigonometric identities? i) ii) iii) iv) v)

2 LESSON SIX- Trigonometric Identities I Example The Pythagorean Identities. a) Using the definition of the unit circle, derive the identity sin x + cos x =. Why is sin x + cos x = called a Pythagorean Identity? Pythagorean Identities b) Verify that sin x + cos x = is an identity using i) x = and ii) x =. c) Verify that sin x + cos x = is an identity using a graphing calculator to draw the graph. sin x + cos x = -

3 LESSON SIX - Trigonometric Identities I d) Using the identity sin x + cos x =, derive + cot x = csc x and tan x + = sec x. e) Verify that + cot x = csc x and tan x + = sec x are identities for x =. f) Verify that + cot x = csc x and tan x + = sec x are identities graphically. + cot x = csc x tan x + = sec x

4 LESSON SIX- Trigonometric Identities I a) Example Prove that each trigonometric statement is an identity. State the non-permissible values of x so the identity is true. Reciprocal Identities NOTE: You will need to use a graphing calculator to obtain the graphs in this lesson. Make sure the calculator is in RADIAN mode, and use window settings that match the grid provided in each example b) -

5 LESSON SIX - Trigonometric Identities I a) Example 4 Prove that each trigonometric statement is an identity. State the non-permissible values of x so the identity is true. Reciprocal Identities b)

6 LESSON SIX- Trigonometric Identities I a) Example 5 Prove that each trigonometric statement is an identity. State the non-permissible values of x so the identity is true. Pythagorean Identities - b) -

7 LESSON SIX - Trigonometric Identities I c) Pythagorean Identities - d) -

8 LESSON SIX- Trigonometric Identities I a) Example 6 Prove that each trigonometric statement is an identity. State the non-permissible values of x so the identity is true. Pythagorean Identities - b) -

9 LESSON SIX - Trigonometric Identities I c) Pythagorean Identities d) 0 -

10 LESSON SIX- Trigonometric Identities I a) Example 7 Prove that each trigonometric statement is an identity. State the non-permissible values of x so the identity is true. Common Denominator Proofs b)

11 LESSON SIX - Trigonometric Identities I c) Common Denominator Proofs d)

12 LESSON SIX- Trigonometric Identities I a) Example 8 Prove that each trigonometric statement is an identity. State the non-permissible values of x so the identity is true. Common Denominator Proofs - - b)

13 LESSON SIX - Trigonometric Identities I c) Common Denominator Proofs d)

14 LESSON SIX- Trigonometric Identities I Example 9 Prove each identity. For simplicity, ignore NPV s and graphs. Assorted Proofs a) b) c) d)

15 LESSON SIX - Trigonometric Identities I Example 0 Prove each identity. For simplicity, ignore NPV s and graphs. Assorted Proofs a) b) c) d)

16 LESSON SIX- Trigonometric Identities I Example Prove each identity. For simplicity, ignore NPV s and graphs. Assorted Proofs a) b) c) d)

17 LESSON SIX - Trigonometric Identities I Example Exploring the proof of Exploring a Proof a) Prove algebraically that b) Verify that for.. c) State the non-permissible values for. d) Show graphically that Are the graphs exactly the same? y = sinx - y = tanxcosx -

18 LESSON SIX- Trigonometric Identities I Example Exploring the proof of Exploring a Proof a) Prove algebraically that b) Verify that for.. c) State the non-permissible values d) Show graphically that for. Are the graphs exactly the same? y = y =

19 LESSON SIX - Trigonometric Identities I Example 4 Exploring the proof of Exploring a Proof a) Prove algebraically that b) Verify that for.. d) Show graphically that c) State the the non-permissible values for. Are the graphs exactly the same? y = y =

20 LESSON SIX- Trigonometric Identities I Example 5 Solve each trigonometric equation over the domain 0 x. Equations With Identities a) b) c) d)

21 LESSON SIX - Trigonometric Identities I Example 6 Solve each trigonometric equation over the domain 0 x. Equations With Identities a) b) c) d) - -

22 LESSON SIX- Trigonometric Identities I Example 7 Solve each trigonometric equation over the domain 0 x. Equations With Identities a) b) c) d)

23 LESSON SIX - Trigonometric Identities I Example 8 Use the Pythagorean identities to find the indicated value and draw the corresponding triangle. Pythagorean Identities and Finding an Unknown a) If the value of find the value of cosx within the same domain. b) If the value of, find the value of seca within the same domain. 7 c) If cosθ =, and cotθ < 0, find the exact value of sinθ. 7

24 LESSON SIX- Trigonometric Identities I Example 9 Trigonometric Substitution. Trigonometric Substitution a) Using the triangle to the right, show that can be expressed as. Hint: Use the triangle to find a trigonometric expression equivalent to b. θ a b b) Using the triangle to the right, show that can be expressed as. Hint: Use the triangle to find a trigonometric expression equivalent to a. θ a 4