Download Trigonometry

Practice Questions:
1.
a) Determine the exact value of:
 
cos 
 12 
2.
Given sin x 
following:
a)
b) Determine an expression that is equivalent to:
 7 
sin 

 6 
5

3 3
 x  2 , determine the exact values for the
, 0  x  and cos y  ,
13
2
7 2
cos 2x
b)
sin x  y 
c)sin2x
d) tan 2x
3.
a)
Solve. Determine exact solutions when possible and approximate solutions otherwise.
2 cos 2 x  5 cos x  3
b)
c)
5 sin x  3  3 sin x
cot x  2.8
4.
Prove.
sec x sin x

 2 tan x
csc x cos x
a)
c)
b)
tan 2 x  cos 2 x 
1
 1  cos 2 x
2
cos x
sin 2 x
 2 csc 2 x  tan x
1  cos 2 x
5.
The population of a ski resort town, as a function of the number of months into a year, can be
described by a cosine function. The maximum population of the town is about 15 000 people, and
the minimum population of the town is about 500 people. At the beginning of the year, the
population is at it’s greatest. After 6 months the population reaches its lowest number of people.
a)
What is the equation of the cosine function that describes the population of this town?
b)
Use the function to find the time(s) at which the population will be 4000.
6.




 3

 x
Simplify sin   x   cos  x   tan   x   tan 
2

2

 2

7.
Prove.
a)
cosx  y cosx  y   cos2 x  cos 2 y  1
8.
Find all solutions in the interval 0  x  2 :
b)
log 10 csc x    log 10 sin x 
a) cos 2x  sin x  1
b) sin 2x sin x  cos x  0
9.
The paddle wheel of a steamboat is 22 feet in diameter and is turning at 3 revolutions per minute.
The axle of the wheel is 8 feet above the surface of the water. Assume that a point P on the edge of
the paddle wheel is at 0,19 at time t  0 seconds.
a)
b)
c)
d)
e)
10.
11.
What maximum height above the water does point P reach?
How far below the water does point P reach?
In how many seconds does the wheel complete one revolution?
Write a cosine function for the height of point P at time t.
Use the function from part d above to find the time(s) at which point P will be at a height
of 10 feet.
State the degree and radian measure of an angle in standard
7
position formed by rotating the terminal side by
of a circle.
12
7
Convert
to degrees.
15
____________
____________
12.
Convert 252 to radians. Write your answer in terms of  .
____________
13.
State the radian measure of one positive and one negative angle in
7
standard position that are co-terminal with
in standard position.
5
____________
14.Determine the length of the circular arc with a central angle of 2 radians.
Assume that the circle has diameter 10.
____________
____________
15.
P (33,56) determines an angle in standard position. Determine all six trigonometric ratios of
angle  .
16.
State the exact value of each of the following.
a)
cos
 5
3
____________
b)
csc
7
4
____________
17.
18.
19.
 2
Write cos
 3

 2
 cos   sin 

 3

 sin  as a single real number.

For y  sec  ,
a)
Determine the location of any
vertical asymptotes.
b)
Sketch the graph for  2    2 .
c)
state the domain and range
For
y  cos X
y  4 cos3x     1
a)
Sketch the graphs for  2    2 .
b)
State the length of the period.
c)
state the range for each graph.
____________