BLC 190
Name:
Worksheet 5
1. Use the unit circle to evaluate the following trigonometric functions. Give exact answers.
(π )
( )
5π
(a) cos
(d) sin
6
6
(b) tan
(π )
(
(c) sin
(e) csc
3
9π
4
(
)
(
(f) cos
4π
3
7π
3
)
)
1
2. If cos θ = , what are the possible values for θ? Use the unit circle, and give the exact
2
answers.
3. If tan θ = 1, what are the possible values for θ? Use the unit circle, and give the exact
answers.
4. The following circle has radius 1 and is centered at the origin:
Ð!Þ)ß !Þ'Ñ
)
(a) What is sin θ?
(b) What is cos θ?
(c) What is tan θ?
5. The following circle has radius 1 and is centered at the origin:
T
)
If the y-coordinate of the point P is 0.5, what is cos θ?
2
6. Sketch the graphs of the following functions:
(a) y = −4 sin x + 3
C
B
(b) y = 3 cos(2x)
C
B
3
7. Find all solutions to the following equations. Give the exact answers in radians.
(a) 2 sin θ + 1 = 0
(b) 3 cos θ = cos θ + 1
(c)
tan θ
1
=
sec θ
2
4
In problems 8 and 9, use the following trig identities:
cos2 θ + sin2 θ = 1
sin2 θ =
1 1
− cos 2θ
2 2
sec2 θ − tan2 θ = 1
cos2 θ =
1 1
+ cos 2θ
2 2
csc2 θ − cot2 θ = 1
sin 2θ = 2 sin θ cos θ
8. Show that (sin x + cos x)2 = 1 + sin 2x.
9. Show that sin2 x + cos 2x = cos2 x.
5
10. Suppose that the following function is used to model the outside temperature on a
certain day of the week:
(
)
π
2π
T (t) = 50 + 10 sin
t−
12
3
where t is the number of hours past midnight, the temperature is measured in Fahrenheit, and radians are used to evaluate sine.
(a) What is the temperature at 8am?
(b) At what time(s) does the temperature equal 45 ◦ F?
(c) Find the minimum temperature.
(d) Find the maximum temperature.
6