Mathematical Investigations: A Collaborative Approach to Understanding Precalculus
Key
Name:
Trigonometry - Beyond the Right Triangles
TO REFRESH THE MEMORY
1.
Evaluate each of the following (without calculators):
 7  1
5 
1
3
a.
sin  =
b.
cot  = 

 6  2
 3 
3
3
  
c.
tan  =  3
d.
cos (1996)= 1
 3 
e.
2.
sin (15°)= sin  45  30   sin 45 cos 30  cos 45 sin 30 
2
2
 23 
2
2
3
12

If cos(u) =  , where  < u < 2, and sin (v) = , where < v < ,
5
13
2
determine exact values for each of the following (without calculators).
a.
cos (u – v) = cos u cos v  sin u sin v    53   135     54  12
13 

15  48
33

65
65
6 2
4
 12 
u
3
4
5
13
12
v
3.
 
tan v  tan u
15 36  20
16

 

12
4
1  tan v tan u 1    5  3  15 15  48
63
b.
tan (v + u) =
c.
2u 
2u 
cos2   + sin2   = 1
 3 
 3 
12
5
4
3
5
Evaluate (without calculators). [NOTE: arcsin(x) = sin-1 (x)]
 1


a.
arcsin   = 
b.
arctan (–1)= 
 2
6
4
c.

 4 
5
sec  arccos     = 
4
 5 

d.
arcsin(sin 210°)= 30

© 2005 Illinois Mathematics and Science Academy® Trig. 13.1
Rev. S06
Mathematical Investigations: A Collaborative Approach to Understanding Precalculus
Key
Name:
4.
Given that d = 12 and B = 24 in BFD , find the values of side b so that:
a.
one triangle exists
b  12 or
b  12sin 24  4.88
b.
c
the triangle does not exist
b  12sin 24
two triangles exist
12sin 24  b  12
T
5.
Solve THE given that H = 25°, e = 41, and h = 25.
sin 25 sin E

25
41
6.
 sin E 
41
25
25
41sin 25
25
H
25
E2
E1
mE1  4353
mE2  180  E1  13607
mT1  180  25  E1  11107
mT2  180  25  E2  1853
sin 25 sin T1
sin 25 sin T2

 t1  55.2

 t2  19.1
25
t1
25
t2
Two sides of a triangle-shaped plot measure 70 m and 122 m. If the angle between these
two sides is 102°, find the area of the plot.
K  12  70122 sin102  4176.7 m2
7.
Two trains leave a station at the same time. One travels due south at 64 km/hour, and the
other travels northeast at 88 km/hour. In how many minutes after they leave will they be
150 km apart?
2
2
 64t   88t 
 64t   88t 
150  
 
  2

 cos135
 60   60 
 60   60 
88t
60
2
  64  2  88  2
2 
 64   88  
22500  t        2      

  60   60 
 60   60   2  

t  63.95 minutes
2
© 2005 Illinois Mathematics and Science Academy® Trig. 13.2
135
150
64t
60
Rev. S06
Mathematical Investigations: A Collaborative Approach to Understanding Precalculus
2 2
33
Name:
sin   
3
7
2
1
4
Given that sin  =  and that cos  = ,
1
3
7
a.
find the largest possible value for cos( + ).
Key
cos   
8.
2
3

2 2

3
1
7


33
4
7
2 2 4  1 
33 
      

3 7  3 
7 
This will be maximized when both terms are positive, so we want
2 2 4 1 33 8 2  33

  

(which happens when   QIV and   QI )
3 7 3 7
21
cos      cos  cos   sin  sin   
b.
find the smallest possible value for sin( – ).
1 4  2 2 
33  4  2 66
sin      sin  cos   cos  sin       




3 7 
3 
7 
21
This will be minimized when the second term is negative, so we want
4  2 66

(which happens when   QIV and   QI or   QIII and   QIV )
21
9.
Sketch each of the following function carefully, labeling important points.
1
1
a.
y = sin 1x  31
b.
y = cos x  1
2
 2, 2 1
 0, 2 
 2, 0 
 4,  2 1
10
Find the angle of inclination for the line y = –2x – 17.
Slope  tan   2    tan 1  2  0
 of incliniation  180    116.57
11.
a.
Find the tangent of an angle between the two lines: y = 3x + 4 and y = –x – 2
tan   3
3 1
4
tan     

 2
tan   1
1  3  1 2
b.
Find an angle between the two lines given in part a.
63.43 and 116.57
© 2005 Illinois Mathematics and Science Academy® Trig. 13.3
Rev. S06
 33
Mathematical Investigations: A Collaborative Approach to Understanding Precalculus
Key
Name:
11.
12.
13.
Simplify without a calculator: (Then check with a calculator.)
a.
cos22.5sin 67.5 sin 22.5cos67.5 = sin  67.5  22.5   sin 45 
b.
cos22.5sin 67.5  sin 22.5cos67.5 = sin  67.5  22.5  sin 90  1
If the point (4, –3) is on the terminal side of angle , find exact values for each of the
following:
a.
sin  = 
3
5
b.
cos  =
4
5
c.
tan  = 
3
4
d.
sin (2) = 2sin  cos   
24
25
Two observers, standing 100 m apart, site a UFO at the same time. The UFO appears to lie
between them. From the first observer, the UFO has an angle of elevation of 78°30' and
from the second, 83°15'. What is the height of the UFO above the ground?
U
mU  180  78.5  83.25  18.25
sin 83.25 sin18.25
100sin 83.25

 s
s
100
sin18.25
h
100sin 83.25 sin 78.5
sin 78.5 
 h  s sin 78.5 
 310.75 m
s
sin18.25
14.
2
2
s
F
h
78.5
f
83.25
u  100 m
7
and sin  < 0, find exact values for each of the following
25
7
24
cos (–) 
b.
sin ( + ) 
25
25
If cos  =
a.
24
7
c.
tan ()  
e.
cos ( + )  
d.
sin ( – )  
24
25
7
25
© 2005 Illinois Mathematics and Science Academy® Trig. 13.4
Rev. S06
S
Mathematical Investigations: A Collaborative Approach to Understanding Precalculus
Key
Name:
15.
16.
Find exact values, assuming x is in the appropriate domain:
 1 3 4
 1 4 3
a.
cos sin   
b.
tan cos   
5 5
5 4


c.
sin(sin-1 (0.2))  0.2
d.
sin-1 (sin 3)  0
e.
sin (cos-1 3x)  1  9x2
f.
 1 x 
16  x 2
tan cos   
4

x
Solve for all values of x (in radians):
2 cos x  1  0
a.
b.
cos x  
2
2
x  4   4  k
3
x
 2k , k 
4
c.
tan2 (x + 4) = 1
cos  x  4   1
x

4
 4  k , k 
sin2 x – 2 sin x + 1 = 0
 sin x  1
2
0
sin x  1

 2 k , k 
2
17. Solve for all values of x if 0  x  2:
a. cos2 4x + cos 4x = 0
cos  4 x   cos  4 x   1  0
x
b.
sin  x  4   1
cos  4 x   0 or cos  4 x   1
x  4   2  2k
4 x  2  k  ,   2 k

x
x
c.
  2 k
8
,
x   2  4  2k
  2 k
4
x
 3 5 7 9 11 13 15
, , ,
, ,
,
,
,
8 8 8 8 8 8
8
8
 3 5 7
, , ,
4 4 4 4
sin(2x) cos x – cos(2x) sin x = 1
d.
sin  2 x  x   sin x  1
x
sin2 (x + 4) = 1

2
© 2005 Illinois Mathematics and Science Academy® Trig. 13.5
3
5
 4,
4
2
2
tan2 x –

3 tan x = 0

tan x tan x  3  0
tan x  0 or tan x  3
 4
x  0,  , 2 , ,
3 3
Rev. S06
Mathematical Investigations: A Collaborative Approach to Understanding Precalculus
Name:
18.
Key
A triangular lot is bounded by two streets.
Find the area, in acres, of the lot. [Note:
1 acre = 43,560 ft2.]
K 
1
2
195 255 sin  4845
 18692.62 ft 2
 0.43 acres
19.
Simplify each of the following expressions.
tan   cot 
a.
b.
csc 

sin 
cos 

 cos
sin  sin 

1
sin 
sin 
sin 2  cos 2 

cos  cos 
1

 sec 
cos 

20.
sin 
1  cos

1  cos
sin 
sin  1  cos   1  cos 


1  cos 2 
sin 
sin  1  cos   1  cos 


sin 2 
sin 
1  cos   1  cos 

sin 
2

 2 csc 
sin 
Prove each of the following identities.
1  sin x
 2 sec 2 x  2 sec x tan x  1
a.
1  sin x
1  sin x 1  sin x 1  sin x 1  2sin x  sin 2 x
1
2sin x sin 2 x
LHS 






1  sin x 1  sin x 1  sin x
1  sin 2 x
cos 2 x cos 2 x cos 2 x
 sec 2 x  2sec x tan x  tan 2 x  sec 2 x  2sec x tan x   sec 2 x  1
 2sec 2 x  2sec x tan x  1  RHS
b.
cos(4 )  1  8sin 2 ( )cos2 ( )
(Note: The cute little  is called "phi.")
LHS  cos  4   cos  2  2   1  2sin  2   1  2  2sin  cos  
2
2
 1  2  4sin 2  cos 2    1  8sin 2  cos 2   RHS
© 2005 Illinois Mathematics and Science Academy® Trig. 13.6
Rev. S06