Mathematical Investigations IV
Name:
Mathematical Investigations IV
Trigonometry - Beyond the Right Triangles
To Refresh the Memory
1.
Evaluate each of the following (without calculators):
 7 
 5 
a.
sin  
b.
cot   =
=
 6 
 3 
 
c.
tan    =
d.
cos (1996)=
 3
e.
2.
3.
sin (15°)=
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) =
b.
tan (v + u) =
c.
 2u 
 2u 
cos2   + sin2   =
 3 
 3 
Evaluate (without calculators). [NOTE: arcsin(x) = sin-1 (x)]
 1
a.
arcsin    =
b.
arctan (–1)=
 2
c.

 4 
sec  arccos     =
 5 

d.
Trig. 13.1
arcsin(sin 210°)=
Rev. F08
Mathematical Investigations IV
Name:
4.
Given that d = 12 and B = 24 in BFD , find the values of side b so that:
a.
one triangle exists
c
the triangle does not exist
b.
two triangles exist
5.
Solve THE given that H = 25°, e = 41, and h = 25.
6.
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.
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?
Trig. 13.2
Rev. F08
Mathematical Investigations IV
Name:
8.
a.
1
4
and that cos  = ,
3
7
find the largest possible value for cos     .
b.
find the smallest possible value for sin     .
Given that sin  = 
9.
Sketch each of the following function carefully, labeling important points.
1
a.
y = sin 1  x  3 1
b.
y = cos 1  x  1
2
10.
Find the angle of inclination for the line y = –2x – 17.
11.
a.
Find the tangent of an angle between (formed by) the two lines:
y  3x  4 and y   x  2 .
b.
Find an angle between the two lines given in part a.
Trig. 13.3
Rev. F08
Mathematical Investigations IV
Name:
12.
13.
Simplify without a calculator: (Then check with a calculator.)
a.
cos 22.5 sin 67.5  sin 22.5 cos67.5 =
b.
cos 22.5 sin 67.5  sin 22.5 cos67.5 =
If the point (4, –3) is on the terminal side of angle , find exact values for each of the
following:
a.
sin  =
b.
cos  =
c.
tan  =
d.
sin (2) =
14.
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?
15.
If cos  =
a.
7
and sin  < 0, find exact values for each of the following
25
cos (–)
b.
sin ( + )
c.
tan ()
e.
cos ( + )
d.
Trig. 13.4
sin ( – )
Rev. F08
Mathematical Investigations IV
Name:
16.
17.
Find exact values, assuming x is in the appropriate domain:


 3 
 4 
a.
cos  sin 1   
b.
tan  cos 1   
 5 
 5 


c.
sin(sin-1 (0.2))
d.
sin-1 (sin 3)
e.
sin (cos-1 3x)
f.

 x 
tan  cos 1   
 4 

b.
tan2 (x + 4) = 1
Solve for all values of x if 0  x  2:
a.
cos2 4x + cos 4x = 0
b.
sin2 (x + 4) = 1
sin(2x) cos x – cos(2x) sin x = 1
d.
tan2 x – 3 tan x = 0
Solve for all values of x (in radians):
a.
2 cos x  1  0
c.
18.
c.
sin2 x – 2 sin x + 1 = 0
Trig. 13.5
Rev. F08
Mathematical Investigations IV
Name:
19.
A triangular lot is bounded by two streets.
Find the area, in acres, of the lot. [Note:
1 acre = 43,560 ft2.]
20.
Simplify each of the following expressions.
tan   cot 
a.
b.
csc 
21.
Prove each of the following identities.
1  sin x
 2sec 2 x  2sec x tan x  1
a.
1  sin x
b.
cos(4 )  1  8sin 2 ( ) cos 2 ( )
sin 
1  cos 

1  cos 
sin 
(Note: The cute little  is called "phi.")
Trig. 13.6
Rev. F08