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Name: _________________________ 9.1B: Pre-Calculus Academic Practice
Simplify each:
1) sin t  cot t cos t
2) sin x cos 2 x  sin x
sec2 x  1
sin 2 x
4) csc  tan   sec 
5) tan x csc x
6) csc x 1  cos 2 x 
7) sin  sec  cos  csc
cos 2 x
8)
1  sin x
sec2 x
9) tan x 
tan x
cos 2 x  4
10)
cos x  2
3)
Prove each identity. Show all steps in your proof.
11) tan x cos x  sin x
12) tan   cot   sec csc
13) cos x sec x  1
15)
1
1
2sin 


1  sin  1  sin 
cos 2 
14)
cot 2 x  tan 2 x
 csc x sec x
cot x tan x
Name: ______________________________________________
9.2A Sum and Difference Trigonometric Identities
Write the expression using a single trigonometric function and angle.
 2
1. cos   cos 
 6
 3
2. sin 
 4

 2 
  sin   sin 


 7 

 2
 cos 

 3

 2
  sin 

 3

 3 
 cos  

 4 
 
 3 
tan    tan  
6
 4 
3.
    3 
1  tan   tan  
6  4 
 2 
 3 
tan 
  tan  
 5 
 2 
4.
 2   3 
1  tan 
 tan  
 5   2 
5. sin  200  cos  600   sin  600  cos  200 
6. cos 1120  cos 190   sin 1120  sin 190 
 
 5 
    5 
7. cos   cos    sin   sin  
3
 6 
3  6 
8.
tan  300   tan  500 
1  tan  300  tan  500 
5
3
1

9. Use cos    ;    
and tan   ;0   
7
2
3
2
a) Find the exact value of sin    
b) Find the exact value of cos    
c) Find the exact value of sin    
d) Find the exact value of cos    
e) Find the exact value of tan    
Name: ______________________________________________
9.2B Sum and Difference Trigonometric Identities
Find the exact value using either sum or difference trigonometric identities
1.
 7 
tan 

 12 
 
2. cos  
 12 
 23 
3. sin 

 12 
 13 
4. cos 

 12 
 17 
5. tan 

 12 
6. cos 150 
7. sin 1650 
8. If sin  
2
12
, cos    and both  ,  are in quadrant II, find
5
13
a. tan    
b. sin    
c. cos    
Name: ______________________________ Academic Practice 9.3A: Pre-Calculus
Find sin (2x), cos (2x), and tan (2x) under the given conditions.
1) sin x 
5 
,  x 
13 2
3 
3) tan x   ,  x  
2 2
5) Simplify
1  cos(2 x)
sin(2 x)
4 3
 x  2
2) sin x   ,
5 2
4) cos x  
8
3
,  x 
17
2
6) Solve sin (2x) + 2 cos x = 0
7) Find the error, correct it, and explain what they did wrong.
__________________________________________
cos 2 x cos 2 x  sin 2 x

cos 2 x
cos 2 x
1

cos 2 x
 sec 2 x
__________________________________________
__________________________________________
__________________________________________
Name: _________________________ Pre-Calculus Academic Practice 9.3B
Use the half-angle identities to find the exact values.
1) sin x, cos x, tan x if x = 75o.
2) sin x, cos x, tan x if x 
17
12
 x
 x
 x
3 & 4: Find sin   , cos   , tan   under the given conditions.
2
2
2
7
3
4

3) sin x   ,   x 
4) tan x  , 0  x 
25
2
3
2
Determine if each is true or false if x is in quadrant III.
x
x
5) sin   is negative __________
6) cos   is negative __________
2
2
x
7) tan   is negative __________
2
x
8) cot   is positive __________
2
1  11 
 11 
9) Fill-in the blanks sin 
 = sin 

2  __ 
 12 
__
1
__
= _________________________
 __
2
 __
1  cos____
2