Proving Trigonometric Identities

Transcription

1 MHF 4UI Unit 7 Day Proving Trigonometric Identities An identity is an epression which is true for all values in the domain. Reciprocal Identities csc θ sin θ sec θ cos θ cot θ tan θ Quotient Identities sin θ tan θ cos θ cos θ cot θ sin θ Pythagorean Identities sin θ cos θ cot θ csc θ tan θ sec θ. Prove each of the following identities. a) cot sin = cos b) ( cos )(csc) = sin

2 MHF 4UI Unit 7 Day c) tan θ - sin θ d) sec θ sin θ cos θ sec θ

3 MHF 4UI Unit 7 Day More Trigonometric Identities Prove each of the following identities. sec a) csc tan sin b) sec - tan sec tan

4 MHF 4UI Unit 7 Day 3 More Trigonometric Identities and Equations A. Addition and Subtraction Trig Identities sin (A + B) = sina cosb + cosa sinb sin (A B) = sina cosb cosa sinb cos(a + B) = cosa cosb sina sinb cos(a B) = cosa cosb + sina sinb tana tanb tan(a B) - tana tanb tana - tanb tan(a - B) tana tanb B. Trig Identities Epand, then determine the eact value. π π a) sin( - ) 4 6 π π b) tan( ) 6 3

5 MHF 4UI Unit 7 Day 3 C. Trig Equations Solve for, where [0, π] a) sin cos π 6 - cos sin π 6 π b) cos cos - sin sin 4 π 4

6 MHF 4UI Unit 7 Day 3 c) sin cos cos sin - 3

7 MHF 4UI Unit 7 Day 4 Investigation: Equivalent Trigonometric Epressions Cofunction Identities In right ABC, B is the right angle. C Mark B on the diagram as. Mark A as on the diagram. The angles in a triangle sum to radians. C = = A Mark this epression for C on the diagram. Complete the following chart (two entries are done for you): b c a B a sin csc sin b cos sec cos tan cot tan csc sec a cot c You should notice that sin and cos both equal a b. sin cos. This equivalent epression is called a cofunction identity. Complete the following chart by discovering all si cofunction identities from the chart above. The first one has been completed for you. Cofunction Identities sin cos cos tan cot csc sec Are these cofunction identities true in general? In the first part of the investigation, the angle would be more than 0 radians but less than radians. i.e. 0,.

8 MHF 4UI Unit 7 Day 4 On the grid sketch y cos. On the grid sketch y sin. y y Sketch y sin using transformations: First rewrite in the form y sin k p What transformations must be applied to y sin to obtain y sin?. reflect in the. translate units to the Apply the reflection to y sin and sketch. y Sketch y cos using transformations: First rewrite in the form y cos k p What transformations must be applied to y cos to obtain y cos? 3. reflect in the 4. translate units to the Apply the reflection to y cos and sketch. y Draw a second sketch by applying the translation to the first sketch. y Draw a second sketch by applying the translation to the first sketch. y Continue another period to the right. Continue another period to the right. What do you notice about the graphs of y cos and y sin? What do you notice about the graphs of y sin and y cos? Write epressions relating the two functions in each of the two columns.

9 y y MHF 4UI Unit 7 Day 4 We could repeat the eercise for each of the si cofunction identities. However, once we have established the first two cofunction identities we can use them algebraically to establish the other four cofunction identities. The first one will be completed for you with appropriate comments. Prove: csc sec Proof: L.S. = csc R.S. = sec = Rewrite csc in terms of sin sin = cos = sec L.S. R.S. csc sec Rewrite But sin using the cofunction identity sec cos, so we can substitute and obtain the required result. Now, using a similar approach to the one shown above, complete the following proofs. Prove: sec csc Prove: cot tan Prove: tan cot Eample: 5 Epress sin in terms Verify this result using special angles and the CAST 6 rule. Do not use a calculator. of the cofunction identity. 5 5 sin cos cos 6 6 cos 6 cos 3 5 sin cos RAA LS sin 6 sin 6 RAA 3 RS cos 3 cos 3 LS RS 5 sin cos 6 3

10 MHF 4UI Unit 7 Day 5 Double Angle Trigonometric Identities and Equations A. Double Angle Trig Identities sin (A) = sinacosa cos(a) = cos A sin A OR cos(a) = sin A OR cos(a) = cos A - tana tan(a) ; where tana - tan A B. Trig Identities Epress as a single sine or cosine function. b) 6 sin cos b) 3 sin(8) cos(8) c) - sin (7θ) d) 8 cos (0 θ) - 9

11 MHF 4UI Unit 7 Day 5 C. Trig Equations Solve for, where [0, π] a) sin = -sin b) cos() sin = 0 c) cos(4) 3cos() = 4

12 MHF 4UI Unit 7 Day 5 Addition Identities Subtraction Identities Double-Angle Identities sin A B sina cosb cosa sinb sin A B sina cosb cosa sinb sin A sinacosa cos A B cosa cosb sina sinb cos A B cosa cosb sina sinb cos A cos A sin A sin A cos A tana tanb tana tanb tana tan( AB) tan( AB) tan( A) tanatan B tanatanb tan A

13 MHF 4UI Unit 7 Day 6 Even More Trigonometric Identities Prove the following identities. a) cos( + y) cos( y) = -sin siny b) sin(π ) = sin

14 MHF 4UI Unit 7 Day 6 c) sin(-) = -sin d) tana tanb tan(a B) - tanatanb

15 MHF 4UI Unit 7 Day 7 Even More Trigonometric Equations Solve for, where [0, π]. State eact answers when possible, otherwise state answers accurate to three decimal places. a) cos(3) b) 3 cos() - sin

16 MHF 4UI Unit 7 Day 7 c) sin() 3 cos() 0

17 MHF 4UI Unit 7 Day 8 Yet Even More Trigonometric Equations Solve for, where [0, π]. State eact answers when possible, otherwise state answers accurate to three decimal places. a) cos 3 tan 0 b) 3 sin - cos cos

18 MHF 4UI Unit 7 Day 8 c) 8 sin cos -cos()

19 MHF 4UI Unit 7 Day 9 Putting It All Together Solve for, where [0, π]. State eact answers when possible, otherwise state answers accurate to three decimal places. a) sin - sin b) cos() - sin 0 c) 3sin() cos d) sin() 3 cos() e) cos 3 tan 0 f) cos cos π 6 sin sin π 6