Download WS - Unit 4 Packet

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Unit 4 Worksheets
4.1 – Trigonometric Identities
(Work on a separate sheet of paper)
Use trigonometric identities to prove the following trigonometric equations.
7.
csc x tan x  sec x
csc x tan x cos x  1
cos2 Acsc Asec A  cot A
sin  cot  cos  csc
sec  cos  sin tan
cos x(sec x  cos x csc2 x)  csc2 x
(sec x  1)(sec x  1)  tan 2 x
8.
1  tan  
1.
2.
3.
4.
5.
6.
2
 sec2   2 tan 
(cos K  sec K )2  tan 2 K  sin 2 K
1  cos 2 x
 sin x cos x
10.
tan x
1  cot 2 
 cot 2 
11.
2
sec 
9.
12.
13.
csc B cos B

 tan B
cos B sin B
1
1

 2sec D
sec D  tan D sec D  tan D
4.2 – Trigonometric Identities
Use the odd / even properties to rewrite with positive arguments:

1. sin (-13)º =
_________________
3. tan (-135º) =
_________________
2. cos    =
 6
4. sec -  =
5. cot (-35º) =
_________________
6. csc (-3489º) =
7. -sec (-73º) =
_________________
8. -tan    =
 5
9. -sin (-305º) =
_________________
11. -csc (-782º) =
_________________


10. -cos    =
 3
12. cot (-213º) =
__________________
__________________
__________________
__________________
__________________
__________________
Use the cofunction properties to rewrite the following: (positive arguments)
13. sin 25º =
__________________
14. sec 12º =
__________________
15. tan 88º =
__________________
16. csc 46º =
__________________
17. cot 13º =
__________________
18. cos 90º =
__________________
__________________
20. cos
__________________
22. cot
__________________
24. csc

=
4
3
21. tan
=
8
19. sin
23. sec 0 =
Use trigonometric identities to prove the
following trigonometric equations.
tan x(cot x  tan x)  sec2 x
2
26. cos x(sec x  cos x)  sin x
2
2
2
27. cos x  tan x cos x  1
2
28. (1  sin  )(1  sin  )  cos 
2
2
2
4
29. cot A csc A  cot A  cot A
4
4
2
30. sec t  tan t  1  2tan t
25.
sec x sin x
31.

 cot x
sin x cos x
1
1
32.

1
sec 2 x csc 2 x
1
33.
 sec 2 r  sec r tan r
1  sin r
sin x
1  cos x
34.

 2csc x
1  cos x
sin x
2
=
3
12 
=
13
7
=
12
__________________
__________________
___________________
35.
1  sin x
 2sec 2 x  2sec x tan x  1
1  sin x
36.
sin3 z cos2 z  cos2 z sin z  cos4 z sin z
37.
sec  tan  
38.
39.
1
sec  tan 
2
sec x  6 tan x  7 tan x  4

sec2 x  5
tan x  2
3
3
sec B  cos B
 sec2 B  1  cos 2 B
sec B  cos B
40.
(2sin x  3cos x)2  (3sin x  2cos x)2  13
41.
(1  tan x)(1  cot( x))  tan x  cot x
42.
1  sec( x)
  csc x
sin( x)  tan( x)
4.3 – Sum and Difference Identities
Demonstrate that the given property really works by substituting: A =
1.
2.
3.
4.
cos(A – B) = cosAcosB + sinAsinB
cos(A + B) = cosAcosB – sinAsinB
sin(A – B) = sinAcosB – cosAsinB
sin(A + B) = sinAcosB + cosAsinB
5. tan(A – B) =
2

,B=
6
3
tan A  tan B
1  tan A tan B
If  and  are the measures of two first quadrant angles, find the exact value of each function.
7
8
and tan  =
, find cos(  -  )
24
17
13
3
7 if csc  =
and tan  = , find tan(  +  )
5
4
24
15
8 if cos  =
and cot  =
, find sin(  -  )
7
17
6 if sin  =
Prove that each of the following is an identity. Plug the left side in the sum/difference formulas
and simplify.
9. sin(Θ + 60º) – cos(Θ + 30º) = sinΘ
10. sin(Θ + 30º) + cos(Θ + 60º) = cosΘ
Show that the left side = the right side by finding the exact values of the following (solve by
plugging in the left side and right side into the sum/difference formulas):
11. sin 75º = cos 15º
12. sin 120º = cos (-30º)
Simplify using composite argument properties.
 4x 
 3x 
 3x   4x 
13. sin 
 cos 
 sin 
 + cos 

 7 
 7   7 
 7 
14. cos65°cos20° + sin65°sin20°
4.4 – Double Angle Identities
Use the double arguments to rewrite the following:
1. sin 2 = ________________________
2. 2 sin x cos x = _____________________
3. sin 70º = _______________________
4. cos 100º = ________________________
5. tan 28 = _______________________
6. cos2 3w – sin2 3w = _________________
7. tan 49º = _______________________
8.
9. sin x cos x = ___________________
10. 2 cos2 40x – 1 = ___________________
11. 1 – 2 sin2 10x = _________________
12. tan A = _________________________
13.
2 tan15
= ___________________
1  tan 2 15
2 tan 34y
= __________________
1  tan 2 34y
14. 4 sin 58º cos 58º = ______________
Find exact values for sin 2A, cos2A, tan2A, sinA/2, cosA/2, tanA/2.
15.
3
sin A  , QI
5
16.
Verify each identity
18.
sin 2 x
 2sin x
cos x
19.
2sin x cos x
 tan 2 x
cos 2 x
20.
2sin x cos x
 tan 2 x sin 2 x
csc2 x  1
21.
sin 2 x
 tan 3 x
cot x 1  cos 2 x 
2
tan A 
3
, QIV
4
17.
cos A 
6
, QII
7
4.5 – Power-Reducing Identities
Prove the power-reducing Identities:
1  cos 2 x
1. sin 2 x 
2
1  cos 2 x
2. cos 2 x 
2
Rewrite the trig functions with no power greater than 1.
3. sin 4 x
4. cos 3 x
5. sin 3 2x
6. sin 5 x