MHF 4UP Chapter 7 Assignment: Trigonometric Identities and Equations STUDENT NAME: _____________________________________________________________ Due Date: Thursday, May 22nd at the BEGINNING OF CLASS Answer all questions to the best of your ability in the space provided. If you run out of space, continue on the back of the page and clearly indicate where to look for the rest of your solution. 1. For each expression in Column 1, choose the expression from Column 2 that completes the identity. You can do your work on rough paper.

Column 1

Column 2 sin2 x − tanxcos x

_______

(a)

! 2 !cos x 2 _______

(b) !sin(−x) !sec x − 1

sec x 1 _______

(c) 2 !csc x !sec x 2 _______

(d) !tanx !1 + sin x

2 2 2 2 _______

(e) !cos x

!csc x − cot x + sin x

2. A student writes “ 1 + cot2 = csc2 ”. Is this student correct? Explain your reasoning.

3. Another student makes the following claim: “Since !sin2 θ + cos2 θ = 1 , I should be able to also say that !sinθ + cosθ = 1 if I take the square root of both sides.” Is this student correct? Explain your reasoning.

4. (a)

(c)

(e)

Prove the following identities.

tanα = sinα

!secα

(b)

!cotθ + tanθ = secθcscθ

(secθ − tanθ)2 + 1 = 2tanθ secθcscθ − tanθcscθ !

(d)

cscθ + cotθ = cotθcscθ tanθ + sinθ !

2 !(1 + sinx + cos x) = 2(1 + sinx)(1 + cos x)

5.

6.

According to Lambert’s law, the intensity of light from a single source

on a flat surface at point P is given by

2 !I = k cos θ ,

where k is a constant. (a) Write I in terms of the sine function.

(b)

Explain why the maximum value of I occurs when !θ = 0 .

Determine the exact value of each expression. (a) ! !sin(−15 )

(b)

7π 2π 7π 2π cos cos − sin sin 9 9 9 9 !

(c)

2 ! !1 − 2sin 22.5

(d)

π 12 2 π 1− tan 12 ! 2tan

7. (a)

(b)

Use the given information to find the exact value of (i) cos(A + B), (ii) tan(A – B), and (iii) sin(2A).

12 −3 tanA = , cosB = , where A and B are in quadrant III 5 ! 5 !

24 15 cos A = , sinB = , where A is in quadrant IV and B is in quadrant II 25 17 ! !

8. (a)

9. (a)

(c)

Prove the following identities.

cos(x − y) = tanx + coty

! cos xsiny

(b)

3 !sin4x = 4sinxcos x − 9sin xcos x

Solve each equation over the interval [0, !2π ]. Use exact solutions where possible; otherwise, round to two decimal places.

!2sinx + 3 = 4

(b)

!2sec x + 2 2 = sec x +

2

!(cot x − 1)( 3cot x + 1) = 0

(d)

2 !2cos x −

3cos x = 0

(e)

(g)

(i)

2 !3sin x − sinx − 1 = 0

(f)

2 !sec x = 2tanx + 4

2 !4sin x − 3 = 0

(h)

!3sinx = − 5cos x

! 3sec 4x − 2 = 0

(j)

!sinx = sin2x