Download XVII - Trigonometric Functions of Real Numbers

MATHEMATICS 201-009-50
Precalculus
Martin Huard
Fall 2007
XVII – Trigonometric Functions of Real Numbers
1. Find the exact value for the trigonometric function at the given number (Do not use a
calculator!)
a) sin ( 23π )
b) cos 34π
c) cos ( 56π )
d) sin ( 52π )
f) sin ( −43π )
e) sin 116π
i) cos
7π
4
j) cos
m) sec 56π
3π
2
n) cot 34π
g) cos π
π
k) tan 3
o) tan ( −4π )
q) csc 32π
r) sec ( 74π )
s) csc ( 76π )
u) csc −136π
v) cot π
w) sec 53π
h) sin 3π
l) csc π6
p) sec −34π
t) tan π
x) cot 54π
2. Find the value of the six trigonometric functions of θ .
a) sin θ = 135 and θ is in quadrant II.
b) cos θ = −53 and θ is in quadrant III.
c) tan θ = − 73 and cos θ < 0
d) csc θ = 3 and tan θ < 0
e) secθ = 2 and csc θ < 0
and sin θ < 0
f) cot θ = 23
g) sin θ = 0 and secθ = −1
h) cot θ = −1 and cos θ > 0
3. Write the first expression in terms of the second if the terminal point determined by t is in the
given quadrant.
a) sin θ in terms of cos θ if θ in quadrant III.
b) sin θ in terms of tan θ if θ is in quadrant II.
c) cos θ in terms of csc θ if θ is in quadrant IV.
d) tan θ in terms of sin θ if θ is in quadrant IV.
e) tan θ in terms of csc θ if θ is in quadrant III.
f) csc θ in terms of cos θ if θ is in quadrant IV.
g) csc θ in terms of tan θ if θ is in quadrant IV.
h) cot θ in terms of secθ if θ is in quadrant III.
i) secθ in terms of tan θ if θ is in quadrant II.
4. If tan θ =
−3
4
and θ is in quadrant IV, find sin θ + cos θ .
5. If cos θ =
−3
4
and θ is in quadrant II, find cot θ + cscθ .
Math 009
XVII – Trigonometric Functions of Real Numbers
6. Find the exact value of each expression. Do not use a calculator.
a) sin π3 − cos π6
b) tan π7 cot π7
c) csc 23π − tan 56π
d) sin π4 − csc π4
cos 25π
sin 25π
e) sin −23π + cos 23π
f) cot 25π −
g) sin 2 17° + cos 2 17°
cos 3°
i) cot 3° −
sin 3°
h) tan 2 π9 − sec 2 π9
j) tan −3π + cot 116π
7. Find the area of the triangle with sides of length 3cm and 7cm and included angle 56o.
8. Find the area of an equilateral triangle with sides of length 15cm.
9. A triangle has an area of 35cm2. If two adjacent sides have length 21cm and 13cm, what is
the angle between the two sides?
Fall 2007
Martin Huard
2
Math 009
XVII – Trigonometric Functions of Real Numbers
ANSWERS
3
2
1. a)
i)
2
2
− 2
2
b)
j) 0
k)
2
q) -1
r)
5
2. a) sin θ = 13
c) −
3
2
3
l) 2
−2 3
3
m)
3
2
g) -1
h) 0
n) -1
o) -1
p) − 2
f)
u) -2
v) ∃
b) sin θ = − 54
s) - 2
t) 0
13
csc θ = 5
w) 2
x) 1
csc θ = − 54
13
sec θ = − 12
cos θ = − 53
sec θ = − 53
tan θ = − 125
cot θ = − 125
tan θ =
cot θ =
csc θ =
3 58
58
sec θ = −
tan θ = − 73
cot θ = − 73
cos θ =
cot θ = −
g) sin θ = 0
cos θ = −1
tan θ = 0
g) csc θ =
1 + tan 2 θ
tan θ
tan θ = −
cot θ = −2 2
2
4
cos θ = − 2 1313
sec θ = −
13
2
tan θ =
cot θ =
cos θ =
3
2
sec θ = 2
2
2
h) cot θ =
tan θ
cot θ = −1
c) cos θ = −
1 + tan 2 θ
1
f) csc θ = −
csc 2 θ − 1
1
2
3
csc θ = − 2
2
2
tan θ = −1
e) tan θ =
1 − sin 2 θ
sec θ = − 3 42
f) sin θ = −
b) sin θ = −
sin θ
cos θ = − 2 3 2
13
3
cot θ = ∃
3. a) sin θ = − 1 − cos 2 θ
csc θ = 3
csc θ = −
3
3
csc θ = ∃
sec θ = −1
3
4
f) sin θ = − 3 1313
sec θ = 2
1
2
tan θ = − 3
d) tan θ =
58
7
csc θ = − 2 3 3
3
2
4
3
d) sin θ = 13
58
3
cos θ = − 7 5858
e) sin θ = −
5.
−1
2
e)
cos θ = − 12
13
c) sin θ =
4.
d) 1
csc 2 θ − 1
csc θ
1
1 − cos 2 θ
i) sec θ = − 1 + tan 2 θ
sec θ − 1
2
1
5
7
7
6. a) 0
b) 1
c)
3
d)
− 2
2
e)
−1− 3
2
f) 0
g) 1
h) -1
i) 0
j) −2 3
2
7. 8.7 cm
8. 97.4 cm2
9. 14.9o or 165.1o
Fall 2007
Martin Huard
3