MTH 114 EXAM 3 REVIEW TOPICS
INVERSES:
- Know the domain and range of y = arcos x, y=arcsinx and y=arctanx
- Know the graphs and key points of y = arcos x, y=arcsinx and y=arctanx
- Understand that the input and output must be of any trig or inverse trig function.
- Be able to evaluate EXACTLY the value of arctrig(#) where that # is on the unit circle.
REMEMBER, angles MUST be in radian mode – NEVER degrees, and in the correct quadrant
for the range of the arctrig function.
- Be able to approximate the value of arctrig(#) using your calculator.
- Determine if trig(arctrig(#))=# or arctrig(trig(angle)=angle based on the domain and range
- Evaluate trig(arctrig(#)) or arctrig(trig(angle)) EXACTLY.
- Make sure you can do all of the problems from the inverse trig worksheet posted at
http://www.math.msu.edu/~giovanni/mth114/invTrigWorksheet.doc
IDENTITIES:
Make sure you have memorized all of the identities and recognize them in different forms!
Be able to simplify expressions using algebra and trig ID’s
Given the value of one trig x = # and info about a quadrant, find the value of all other trig functions
using trig ID’s only (NO Triangles allowed).
Be able to verify Identities. Remember you MUST WORK ON ONE SIDE ONLY, clearly show all
steps. Make sure you NEVER drop your variables cos is not the same as cos x! Practice problems
using all of the common techniques – using id’s, combining fractions, breaking up fractions,
simplifying complex fractions, conjugates, expanding or combining sum formulas, double angle, halfangle id’s, converting to all sines and cosines etc. Watch out for common algebra mistakes!
Find the exact value of sin, cos or tan of an angle that is the sum or difference of 2 angles on the unit
circle or half an angle on the unit circle. For example evaluate sin(13π/12) or tan(17π/12), cos(π/8)
etc. You may need to find the angles yourself! You will likely be required to simplify expressions
completely. This may require simplifying complex fractions, radical expressions or rationalizing
denominators etc.
Given trig A = # and info about what quadrant A is in, and trig(B) = # and info about the quadrant B is
in, find the EXACT value of
Trig(A+/- B), Trig(-A), Trig(pi/2 – A), Trig(2A), Trig (A/2), a different trig function of A etc. You
may be required to simplify your answers completely. Make sure you remember to draw your
triangles in the appropriate quadrants and analyze the sign values get the proper sign values of the
missing sides. You MUST SHOW the Pythagorean step AND justification for the sign of the missing
side to receive full credit. WATCH CAREFULLY that the legs of your triangle are correct for the
quad you are in and the reference angle and sides are clearly labeled in your picture. When solving
half angle problems, make sure you remember to carefully analyze the sign value and quadrants. The
sign outside the radical comes from the sign in the quad of the angle A/2. You must show the work
identifying what quadrant A/2 is in and why you chose the sign value this to receive full credit. The
sign under the radical is the sign of cos A in the quad of the angle A. Watch sign values
CAREFULLY!
Find the exact value of Trig(U+/- V), Trig(-U), Trig(2U), Trig (U/2) where U and V are of the form
invtrig(#). For example cos((sin 1 ( 34 )  tan 1 2) or sin( 2 tan 1 ( 13 ))
Make sure you can do all of the problems on the multiple angle worksheet posted at
http://www.math.msu.edu/~giovanni/mth114/MultipleAngleWorksheet.doc!
EQUATIONS:
Solving equations: Be able to solve trig equations both in [0, 2pi) and in general. Practice problems
with a variety of techniques – isolating the trig function, factoring, quadratic equation, using trig id’s,
taking square roots (DON”T forget the +/-) etc. You should know how to solve them exactly if the # is
on the unit circle and approximate on a calculator using reference angles if not. Make sure you
understand how to do ALL of the book homework, worksheets graded homework and all examples in
class. Make sure you can solve equations involving angles other than just x! Watch out that you do
not invent trig id or algebra rules that do not exist!
Solve trig( ) = # not on the unit circle. You will need to show all reference angle work!
Make sure you can do all textbook problems, worksheet problems, lecture examples etc.
EXTRA PROBLEMS FOR EXAM 3:
1. Given
cos u 
a) sin u
g) sin(v/2)
2 2
3
, sin v  53 , 32  u  2 , 52  v  3 , find the exact value of
b) cos(-v)
h) cos(u-v)
2. Given tan  
c) sin(u+v)
I) csc(2v)
d) cos(2v)
j) tan(2u)
e) sin(2u)
k) tan(v/2)
f) cos(u/2)
4
7
, 3   
Find the EXACT value of each of the following. Simplify
3
2
completely.
 
b) sin  
2
 
a) cos 
2
 
c) tan  
2
d) tan( 2  )
3. Solve the following trig equations algebraically in [0, 2 ) and in general. Give all answers EXACTLY if possible.
Otherwise give answers accurate to 0.0001.
a) sin
1
( 135 )  cos 1 ( 54 )  sin 1 t
c)2 cos(2x)+1 = 0
f) sin( 2 x ) 
i) csc(2x)+4=0
d)
b) 2 3 sin( 2 x) tan x  3 tan x
4 sin x cos x  3
2 sin x  0
j) 4sin2x -1 = 0
e) 2sin(3x)cot(2x)=cot(2x)
g) 7cos(2x)+3=0 h) 2tan(3x)+1=0
k) 2 cos(3 x) 
3  0 l) (2 sin( 4 x)  1)(3 cos x  2)  0
4. Given
5
2
sin(u/2)
 u  3 . Fill in the following table with exact values
cos(u/2)
Sinu
3/5
Cosu
sin(2u)
cos(2u)
Tan(2u)
-8/17
7/8
-2/3
17
5
17
5
 7 
 7 
 7 
) cos( )  sin(
) sin( )
, cos
 and tan 
. b) cos(
12
12
12
12
 12 
 12 
 12 
5. Find the EXACT value of a) sin 
6. Evaluate the following exactly:
1 4
5
a) cos( 2 sin
)
b) sin( 2 tan
1 x
3
) c) sin(tan
1
1
2
( )  cos 1 ( ))
3
5
2
3
1

) e) sin 1 ( ) f) tan 1 (
) g) cos 1 ( )
2
2
2
3
5
7
24
sin 1 (sin( )) i) cos 1 (cos( )) j) tan( 2 sin 1  ) k) cos(cos 1
3
6
25
1
d) cos ( 
h)
1
7. a) Cos ( 
2
3
)
 3
) = _____
3
4
f) Cos 1 (cos( ))
3
3
2
) = ____ b) Sin 1 ( 
) = ____
2
2
c) Tan 1 (

d) h) Cos 1 ( )  ____ e)) Tan 1 (1) = _____
2
 5
14
))
))
g) Tan 1 (tan(
h) Sin 1 (sin(
4
3
j) Sin 1 (sin( 118 ))
i) Cos 1 (cos( 118 ))
8 Verify the following ID:
a)
4 sin 2x cos 2x  2 sin x
b)
(sin x  cos x) 2  1  sin(2 x)
cos(4t )  8 cos4 t  8 cos2 t  1
4
e) sin t  83  21 cos( 2t )  81 cos(4t )
c)
u
7/4
23/6
16/3
13/8
sin
u
d)
sin(4t )  4 sin t cos t (1  2 sin 2 t )
f)
2 sin2 (2t )  cos(4t )  1
cot( 2 )  tan( 2 )
cot 2 (3 )  1
i) cos(6 ) 
sec( 2 ) csc( 2 )
csc 2 (3 )
csc( 4 ) sec( 4 )
k) 2 cot(8 ) 

sec( 4 ) csc( 4 )
g)
sin(3x)  3 sin x  4 sin3 x h) cos( 4 ) 
j)
cos( 4 ) 
cot( 2 )  tan( 2 )
cot( 2 )  tan( 2 )
9. Fill in the following table with exact values
cos
tan
sin
cos
tan
sin(2u)
u
u
u/2
u/2
u/2
SKIP
SKIP
SKIP
cos(2u)
tan(2u)
sin 1 (sin u)
cos 1 (cos u)
tan 1 (tan u )
1
in [0, 2π)
2
1
1
d) Solve sin( 5 x)   in [0, 2π) e) Find Sin 1 (  )
2
2
10. a) Solve sin x  
1
2
b) Solve sin x  
c) Solve sin( 5 x)  
1
2
Key to exam 3 extra problems:
1. a) -1/3
b) -4/5 c)
2
15
(3 2  2)
d) 7/25 e)  4 9 2
f)

32 2
6
g)  3 1010
8 2
h)  315
i) -25/24 j)  4 7 2 k) 3
2 a)
5
5
b) 
2 5
5
c) -2
d) -24/7
3. a) t=(-5/13)(4/5)+(12/13)(3/5) = 16/65
b) 0, 2/3, 5/6, , 5/3, 11/6 GS: x=0+nπ, x=2π/3+nπ, x=5/6 + nπ
c) /3, 2/3, 4/3, 5/3
GS: x=π/3+nπ, x=2π/3+nπ
d) /6, /3, 7/6, 4/3
GS: x=π/6+nπ, x=π/3+nπ
e) 0, /2, , 3/2, /18, 13/18, 25/18, 5/18, 17/18, 29/18
GS: x=0+nπ/2, x=π/18+2nπ/3, x=5π/18+2nπ/3
f) 0, , 3/4, 5/4
GS: : x=0+nπ, x=3π/4+2nπ, : x=5π/4+2nπ
g) arccos(-3/7)/2 =1.0069, arccos(-3/7)/2 + π=4.1484, (2π-arccos(-3/7))/2=2.1347, (2π-arccos(3/7))/2 + π = 5.2763
GS: ) arccos(-3/7)/2 + nπ =1.0069 + nπ, (2π-arccos(-3/7))/2 + nπ=2.1347 + nπ
h) (2π+arctan(-1/2))/3 =1.9398, (2π+arctan(-1/2))/3 - π/3=.8926, (2π+arctan(-1/2))/3 +
π/3=2.9870, (2π+arctan(-1/2))/3 + 2π/3=4.0342, (2π+arctan(-1/2))/3 + π=5.0814, (2π+arctan(1/2))/3 + 4π/3=6.1286
GS: (2π+arctan(-1/2))/3 + nπ/3
i) arcsin(-1/4)/2 + π=3.0153, arcsin(-1/4)/2 + 2π=6.1568, (π-arcsin(-1/4))/2=1.6971, (π-arcsin(1/4))/2+π=4.8387
GS: ) arcsin(-1/4)/2 + nπ=3.0153 + nπ, (π-arcsin(-1/4))/2+ nπ=1.6971+ nπ
j) /6, 5π/6, 7/6, 11/6
GS: x=π/6+nπ, x=5π/6+nπ
k) /18, 13/18, 25/18, 11/18, 23/18, 35/18
GS: x=π/18+nπ/3, x=11π/18+nπ/3
5 11 17 23 29
;
;
;
;
24 24 24 24 24
5 k
11

;x 
GS: x 
24
2
24
l)
35 41
;
;
24 24
k

;2.3005  2k ;x  3.9827  2k
2
4.
sin(u/2)
-310/10
-534/34
-
8  15
4
-(5/6)
cos(u/2)
-10/10
-334/34
-
8  15
4
-6/6
Sinu
3/5
15/17
7/8
Cosu
-4/5
-8/17
-15/8
sin(2u)
-24/25
-240/289
-715/32
cos(2u)
7/25
-161/289
-17/32
tan(2u)
-24/7
240/161
715/17
5/3
-2/3
-45/9
-1/9
45
11
3
2 3
7
2 3
7
)
; cos

; tan
 2  3 b)  cos(
6
2
2
12
2
12
2 10  3 210
6. a) -7/25
b) x 62 x 9 c)
d) 3/4 e) /6 f) -/6 g) Undefined h)-/3 i) 5/6
50
2
 1 so not in domain of Cos 1 x which is [-1, 1]
j)336/527 k) undefined because
3
5. a) sin
7. a)
7

12
3 5
5
Cos 1 (
)=
2
6
6
b) Sin 1 ( 
2

)= 
2
4


 1 so not in [-1, 1]
d) h) Cos 1 ( )  undefined
2
2
4
2
 5

)) = 
f) Cos 1 (cos( )) =
g) Tan 1 (tan(
3
3
4
4
5

3

i) Cos 1 (cos( 118 )) =
j) Sin 1 (sin( 118 )) = 
8
8
c) Tan 1 (
 3

)= 
3
6
e)) Tan 1 (1) = 
h) Sin 1 (sin(

4
14

)) =
3
3
8. a) 4 sin 2x cos 2x  2(2 sin 2x cos 2x )  2 sin(2 2x )  2 sin x
b) (sin x  cos x) 2  sin2 x  2 sin x cos x  cos2 x  (sin2 x  cos2 x)  2 sin x cos x  1  sin(2 x)
c) cos(4t)  2cos 2 (2t)  1  2(2cos 2 t  1) 2  1
 2(4cos 4 t  4 cost  1)  1  8cos 4 t  8cost  2  1  8cos4 t  8cos t  1
d) sin(4t )  2 sin(2t ) cos(2t )  2(2 sin t cos t )(1  2 sin2 t )  4 sin t cos t (1  2 sin2 t )
2t ) 2
e) sin 4 t  (sin 2 t ) 2  ( 1cos(
)  14 (1  2 cos( 2t )  cos 2 (2t ))
2
 14 (1  2 cos( 2t )  12 (1  cos( 4t ))  14 ( 32  2 cos( 2t )  12 cos( 4t ))
 83  12 cos( 2t )  18 cos( 4t )
f) 2 sin 2 (2t )  cos(4t )  2 sin 2 (2t )  1  2 sin 2 (2t )  1
g ) sin(3x)  sin( x  2 x)  sin x cos(2 x)  cos x sin(2 x)  sin x(1  2 sin2 x)  cos x(2 sin x cos x)
 sin x  2 sin3 x  2 sin x(1  sin2 x)  sin x  2 sin3 x  2 sin x  2 sin3 x  3 sin x  4 sin3 x
cos( 2 ) sin( 2 ) cos( 2 ) sin( 2 )


cot( 2 )  tan( 2 ) sin( 2 ) cos( 2 ) sin( 2 ) cos( 2 ) sin( 2 ) cos( 2 )



1
1
1
1
sec( 2 ) csc( 2 )
sin( 2 ) cos( 2 )


h)
cos( 2 ) sin( 2 )
cos( 2 ) sin( 2 )
cos 2 (2 )  sin 2 (2 )
 cos( 2(2 ))  cos( 4 )
1

i)
cos 2 (3 )
cos 2 (3 )

1
1
cot 2 (3 )  1 sin 2 (3 )
sin 2 (3 ) cos 2 (3 )  sin 2 (3 )
sin 2 (3 )




 cos( 2(3 ))  cos(6 )
1
1
1
csc 2 (3 )
sin 2 (3 )
sin 2 (3 )
sin 2 (3 )
j)
cot( 2 )  tan( 2 )

cot( 2 )  tan( 2 )

k)
cos(2 )
sin(2 )
sin(2 )
 cos(
2 )
cos(2 )
sin(2 )
sin(2 )
 cos(
2 )

cos(2 )
sin(2 )
sin(2 )
 cos(
2 )
cos(2 )
sin(2 )
sin(2 )
 cos(
2 )
sin( 2 ) cos( 2 ) cos 2 (2 )  sin 2 (2 )


sin( 2 ) cos( 2 ) cos 2 (2 )  sin 2 (2 )
cos( 2(2 ))
 cos( 4 )
1
csc( 4 ) sec( 4 )


sec( 4 ) csc( 4 )

1
sin(4 )
1
cos(4 )

1
cos(4 )
1
sin(4 )

cos( 4 ) sin( 4 ) cos 2 (4 )  sin 2 (4 )


sin( 4 ) cos( 4 )
sin( 4 ) cos( 4 )
2(cos 2 (4 )  sin 2 (4 )) 2 cos( 2(4 ))
cos(8 )

2
 2 cot(8 )
2 sin( 4 ) cos( 4 )
sin( 2(4 ))
sin( 8 )
9.
u
sin u
-2/2
cos u
2/2
tan u
-1
sin u/2
7/4
23/6
-1/2
3/2
-1/3
-
16/3
-3/2
-1/2
13/8
 2 2 2
Note:
Must
use ½
angle!
3
-1-2
Note:
Must
use ½
angle!
-1/2
3/2
SKIP
SKIP
Can do
using
answer
from
cosu if
needed!
2 2
2
Note:
Must
use ½
angle!
2 2
2
2 3
2
U
sin 1 (sin u)
cos 1 (cos u)
tan 1 (tan u )
7/4
23/6
16/3
13/8
-π/4
-π/6
-π/3
-3/8
π/4
π/6
2π/3
3/8
-π/4
-π/6
π/3
-3/8
cos u/2

2 2
2
2 3
2
tan u/2
1-2
sin(2u)
-1
cos(2u)
0
tan(2u)
undef
3-2
-3/2
1/2
-3
-3
SKIP
3/2
-2/2
-1/2
-2/2
-3
1
5
11
5
11
 2k
11 2k
 2k ; x 
 2k b) x 
;x 
;x

c) x  
6
6
6
6
6
5
30
5

11
17
23
29
7
41
47
53
59
;x
;x
;x
;x 
;x 
;x 
;x 
;x 
d) x  ; x 
6
30
30
30
30
6
30
30
30
30
10. a) x 
e) 

6
1
1
has 2 answers (plus generalizer for GS), BUT Sin 1 (  ) can ONLY
2
2
1
HAVE 1 and it must be in the range of Sin ( x) . There are 5 times as many solutions for
1
1
sin( 5 x)   in [0, 2π)
as sin x   in [0, 2π)!
2
2
Notice the solution to sin x  