Math 102 Sample Test 3, Spring 2014
2
1. Verify the identity
Name _______________________________
2
tan x − cot x
= csc x sec x
tan x − cot x
Recognize that the left side is (a2 –b2)/(a-b), so it becomes tan x + cot x. Then, put everything in terms of cos and
sin.
sin x/cos x + cos x/sin x = (1/ sinx)( 1/cos x). The get rid of the denominators by multiplying both sides by
cos x sinx . This leaves sin2x+ cos2x = 1, Pythagorean identity.
2. What is the exact value of:
(Hint: Each of these can be evaluated by an identity.)
π 
2  π 
 + sin   ?
 24 
 24 
a.) cos 2 
π 
2 π 
 − sin   ? Cos π/6 Half angle identity
 12 
 12 
1, Pythag b.) cos 2 
( )
3. Find the exact value of sin 75o using a sum or difference formula.
75 = 45 + 30
sin (a+b) = sin a cos b + cos a sin b = sqrt(2)/2[1/2 + sqrt(3)/2] = [sqrt(2)+sqrt(6)]/4
4. Find the exact value of cos (α + β ) , given sin α = −
4
5
, cos β =
, α terminates in quadrant III, and β
5
3
terminates in quadrant IV. Cos(a+b) = cos a cos b – sin a sin b; cos a = -3/5, sin b = - 2/3
cos(a+b) = (-3/5)(sqrt(5)/3) – (-4/5)(-2/3) = -sqrt(5)/5 – 8/15
Find the exact value of sin ( 2θ ) , cos ( 2θ ) , and tan ( 2θ ) using cos θ = −
5.
8
; tan θ > 0 .
17
sin 2x = 2sinx cosx, cos 2x = cos2x - sin2x. If cos x = -8/17, sin x = -15/17. Sign? <0 since tan>0
cos2x = (64 – 225) (17) 2 sin2x = 2 (120)/(17) 2 tan 2x = 240/161
8
θ 
θ 
θ 
 , cos   , and tan   using cos θ = − ; θ in QIII .
17
2
2
2
6. Find the exact value of sin 
sin x/2 = ± sqrt[(1- cos x)/2], cos x/2 = ± sqrt[(1+ cos x)/2]
sin (θ/2) = sqrt[(1+8/17)/2] = +sqrt(25/34), cos(θ/2) = -sqrt[(1- 8/17)/2] = sqrt(9/34); Signs are since are in Q2.
tan(θ/2) = sin (θ/2)/ cos(θ/2) = sqrt(25/9) = -5/3.
7. Evaluate each expression without using a calculator and write your answer using radians.

a. sin −1  −


a. 4π/3, 5π/3
3
 (4 points)
2 

b. cos −1  −


b. 3π/4, 5π/4
2
 (4 points)
2 

c. tan −1  −

c. 2π/3, 5π/3
1 
 (4 points)
3
8. Sketch a graph of y = tan −1 x and state its domain and range.
Domain (-∞, ∞), Range [-π/2, π/2)
1.5
1
Value
0.5
0
-0.5
-1
-1.5
Angle, Radians
9. Evaluate the expressions by drawing a right triangle and labeling its sides.

sin  tan −1

Op = x, adj = sqrt(x2 +4), hyp = sqrt(2x2 +4), sin = x/sqrt(2x2 +4)


x2 + 4 
x
Sqrt(2x2+4)
x
Sqrt(x2+4)
10. Solve (a) for the principal root, (b) for all solutions in the interval [0,2π), and (c) all real roots.
8sin x = −4 2
Sin x = -sqrt(2)/2, x = 5 π/4, 7π/4
Principle: 7π/4
[0,2π): 5 π/4, 7π/4
All: 5 π/4 + 2nπ, 7π/4+ 2nπ
11. Solve (a) for the principal root, (b) for all solutions in the interval [0,2π), and (c) all real roots.
− 3 sec ( 3 x ) = 2
Sec 3x = - 2/sqrt(3), 3x = 2π/3, 4π/3
Principle: 2π/3, in interval [0,2π) 2π/3, 4π/3; all 2π/3 + 2/3 nπ, 4π/3+ 2/3 nπ
12. Solve the equation in [0,2π) using the method indicated
4 sin 3 x − 4sin 2 x − sin x + 1 = 0
(factor by grouping)
4 sin2x(sinx – 1) – (sinx -1) = (4 sin2x -1) (sinx -1); sinx = 1, x = π/2, sin2x = ¼, sinx = ± ½, x = π/6, 5π/6, 7π/6, 11π/6
13. State the period P of each function and find all solutions in [0,P).
π
π
−150 cos  x +  + 75 = 0
6
4
Cos(π/4x + π/6) = ½, π/4 x + π/6 = π/6 or 11 π/6. Remove the π, x/4 = 0 so x = 0 or
x/4 = 5/3, x = 20/3
Both are in the period [0, 8)
P = ___8______________