Download Math 162 Review for Exam 3

Math 162 Review for Exam 3
1. Find the exact values.
(a) tan(15o)
(b) sec(210o)
 
(e) cot 
8
 
(d) cos 
 12 
(c) sin(22.5o)
2. Find the exact values of each of the remaining five trigonometric functions.
2
3
(a) sin  
and  is in quadrant II.
(b) cos    and  is in quadrant III.
3
8
1
3
(c) tan    and  is in quadrant IV.
(d) sec  
and  is in quadrant I.
4
2
4. Simplify.
cos x cot x
1
(a)
1  sin x
(d) cos 3 x  sin 2 x cos x
1
1

1  sin x 1  csc x


(e) cos  x  csc x
2

(b)
(c)
1  sin x
cos x

cos x
1  sin x
(f) 2 sin(17o) cos(17o)
5. Solve for x.
(a) 4 cos x  3  cos 2 x
(d) tan x 
1
2
(b) tan 2 x  5 tan x  14
(e) sec x 
1
2
(c) sin x 
1
2
(f) 1  2 sin 2 x  2 cos 2 x  1
6. Prove that each equation is an identity.
1
1

 tan x  cot x
(a)
tan x cot x
(b) cos x 
(c) cos x  sin x tan x  sec x
(d) sin 2 x  sin 4 x  cos 2 x  cos 4 x
cos x
sin x cos x

1  tan x sin x  cos x
7. The angle of elevation of the top of a mountain is 30o. One mile further from the mountain
the angle of elevation is 20o. How tall is the mountain?
Math 162 Answers to Review 3
1.(a) 2  3 (b) 
(c)
2 2
2
(d)
6 2
4
or
2 3
(e) 1 2
2
2 5
5
5
3 5
3
, tan   
, cot   
, sec   
, csc  
5
2
3
2
5
2.(a) cos   
(b) sin   
(c) sin   
(d) sin  
2 3
3
3 55
8 55
55
55
8
, tan  
, cot  
, sec    , csc   
55
55
3
8
3
17
4 17
17
, cos  
, cot   4 , sec  
, csc    17
17
17
4
5
5
2 5
3 5
2
, cos   , tan  
, cot  
, csc 
2
3
3
5
5
3.(a) b = 3.65762515,  = 43.11803803o,  = 106.881962o
(b) c = 7.927837671,  = 31.88883083o,  = 123.1111692o
or c = 1.135240197,  = 148.1111692o,  = 6.88883083o
(c) a = 7.42227199, b = 3.949308436,  = 120o
(d) not possible
4.(a) csc x (b) 1 (c) 2sec x (d) cos x
(e) 1
(f) sin 34o
5.(a) x = 2 k
(b) x  tan 1  7   k  1.428899272 + k or x  tan 1 2  k  1.107148718 + k
(c) x 

6
 2k or x 
5
 2k
6
1
(d) x  tan 1    k  0.463647609 + k
2
(e) no solution. The equation is a contradiction.
(f) x can be any number. The equation is an identity.
Math 162 Answers to Review 3
6.(a) Left-Hand-Side =
1
1


tan x cot x
1
sin x
cos x
1
cos x
sin x


cos x sin x

sin x cos x
 cot x  tan x  tan x  cot x = Right-Hand-Side
(b) Left-Hand-Side = cos x 
cos x
cos x
cos 2 x
 cos x 
 cos x 
sin x
1  tan x
cos x  sin x
1
cos x
cos x  cos x  sin x 
cos 2 x
cos 2 x  cos x sin x  cos 2 x  sin x cos x




1  cos x  sin x 
cos x  sin x
cos x  sin x
cos x  sin x

sin x cos x
= Right-Hand-Side
sin x  cos x
(c) Left-Hand-Side = cos x  sin x tan x 
cos x sin x sin x cos x sin 2 x



1
1 cos x
1
cos x
cos x  cos x sin 2 x cos 2 x  sin 2 x
1




 sec x = Right-Hand-Side
1  cos x
cos x
cos x
cos x


(d) Left-Hand-Side = sin 2 x  sin 4 x  sin 2 x 1  sin 2 x  sin 2 x cos 2 x


 1  cos 2 x cos 2 x  cos 2 x  cos 4 x = Right-Hand-Side
7. Approximately 0.984807753 miles.