Download Scholarship Precalculus

Good Luck to ___________________________________
Period _____
Scholarship Pre-Calculus Test
Score ________________
(2011-2012)
SHOW CLEAR, VALID, STEP-BY-STEP WORK, or loss of credit.
All ANSWERS must be EXACT and in SIMPLEST FORM, unless otherwise stated.
BOX YOUR FINAL ANSWERS.
Write the expression as the sine, cosine, or tangent of an angle. You must show a work step that
illustrates two angles be added or subtracted, or one angle being doubled or halved. Also, give the
exact value of the expression if the angle obtained is quadrantal or the angle obtained has a special
reference angle.
1.
sin 50 cos 32  cos 50 sin 32 
2.
cos 110 cos 40  sin 110 sin 40 
3.
tan 50  tan 10
1  tan 50 tan 10
4.
2 sin 22.5 cos 22.5 
5.
cos 2
6.
2 cos 2
7.
4 sin 105 cos 105 
8.
2 tan 75
1  tan 2 75
9.

8
 sin 2

8

8

1 
1  cos 50
2



10.
1  cos 80
2

Find the exact value of the expression by using a Sum or Difference Formula.
11.
Find the exact value of sin105 by expanding and evaluating the following expression.
sin105  sin  60  45
12.
Find the exact value of cos165 by expanding and evaluating the following expression.
cos165  cos 120  45
13.
Find the exact value of tan 75 by expanding and evaluating the following expression.
tan 75  tan  45  30
Find the exact value of the expression by using a Double-Angle Formula.
14.
sin x  
1
3
,  x
5
2
.
Find the exact value of sin 2x .
sin 2x 
15.
cos x 
2
,
3
3
 x  2 .
2
a.
sin 2x 
b.
cos 2x 
c.
tan 2x 
Find the exact value of sin 2x , cos 2x , and tan 2x .
Find the exact value of the expression by using a Half-Angle Formula.
16.
Find the exact value of sin 75 by using a Half-Angle Formula.
sin 75 
17.
Find the exact value of cos 67.5 by using a Half-Angle Formula.
cos 67.5 
18.
Find the exact value of tan 75 by using the Half-Angle Formula tan
tan 75 
Find the exact solutions of the equation over the given domain.
19.
sin 2 x  sin x , 0  x  2
x
1  cos x
.

2
sin x
20.
cos 2 x  cos x , 0  x  2
Verify the trigonometric identity by transforming one side into the other side.
21.
sin  x      sin x
sin  x    
22.
 sin x  cos x 
2
 sin x  cos x 
23.
2
 1  sin 2 x

sin 2 x
1  cos 2 x
 cot x
sin 2 x
1  cos 2 x

24.
sin 2 x
cos 2 x

sin x
cos x
 sec x
sin 2 x
cos 2 x

sin x
cos x

Solve each problem.
25.
sin  
4

3
and tan   3 , where
. Find sin     .
    and    
5
2
2
sin     
26.
Derive the following Triple-Angle Formula by expanding and simplifying the given expression.
sin 3x  3sin x  4sin 3 x
Derivation (Proof)
sin 3x  sin  2 x  x 

27.
6 tan 2   4  5tan 
28.
Use the given Product-to-Sum Formula to write the given product as a sum or difference, then
find the exact value of the expression. Give the value as a single fraction.
Given:
sin u cos v 
1
sin  u  v   sin  u  v  
2
sin 75 cos15 
29.
Use the given Sum-to-Product Formula to write the given sum (or difference) as a product,
then find the exact value of the expression. Give the value as a single fraction.
Given:
uv
cos u  cos v  2 cos 
 2
cos195  cos105 

 uv 
 cos 


 2 