Homework Assignment
Solutions
1
3
4.5.1. If cos(θ) = and sin(θ) < 0, find the following and
give exact answers:
(a) sin(θ)
Answer:
(b) tan(θ)
√
Answer: −2 2
√
or −
2
4
(e) csc(θ)
3
Answer: − 2√
or
2
√
−342
4.5.2. If csc(θ) = −3/2 and cos(θ) < 0, find the following and
give exact answers:
(a) sin(θ)
Answer: − 32
(c) tan(θ)
or
or
√
2
(i) cot( 11π
6 )
√
Answer: − 3
Answer: 2.73
Answer: − √35 or
(e) cot(θ)
√
−355
√
Answer: −
4.5.9. Suppose θ is an angle such that csc(θ) = − 73 and
3π
π
2 ≤θ ≤ 2 .
√
Answer: − 2 3 2
(a) Find cos(θ).
(c) tan(θ)
1
or −
Answer: − 2√
3
(d) sec(θ)
3
Answer: − 2√
or
2
√
2
4
√
−342
(e) cot(θ)
√
Answer: −2 2
4.5.4. Calculate the following and give exact answers:
(a) sec( 11π
6 )
√2
3
27
4.5.8. Find an angle θ such that csc(θ) = − 13
, sec(θ) < 0,
and 0 ≤ θ ≤ 2π. Round your answer to two decimal
places.
Answer: 4.01
1
3
(b) cos(θ)
4.5.7. Find an angle θ such that 0 ≤ θ ≤ 2π, sec(θ) = 2, and
csc(θ) < 0. Round to two decimal places.
Answer: 5.24
5
3
4.5.3. If csc(θ) = 3 and π2 ≤ θ ≤ π, find the following and
give exact answers:
Answer:
√2
2
4.5.6. Suppose that 0 ≤ θ ≤ 2π, csc(θ) = 5/2 and sec(θ) < 0.
Find θ and round to two decimal places.
√
2 5
5
(d) sec(θ)
(a) sin(θ)
Answer:
Answer:
Answer: 2.70
5
3
√2
5
(h) csc( 3π
4 )
4.5.5. Find an angle θ such that csc(θ) = 37 and sec(θ) < 0
where 0 ≤ θ ≤ 2π. Round your answer to two decimal
places.
√
Answer: −
Answer:
(f) cot( 5π
4 )
Answer: 1
(g) sec( 7π
3 )
Answer: 2
(d) sec(θ)
Answer: 3
(b) cos(θ)
√
Answer: − √23 or − 2 3 3
(e) csc( 3π
2 )
Answer: −1
(c) cot(θ)
Answer:
(c)
(d) sec( 4π
3 )
Answer: −2
√
−232
1
− 2√
2
Section 4.5
csc( 4π
3 )
or
√
2 3
3
(b) cot( −3π
4 )
Answer: 1
Last Updated: March 29, 2014
Answer: − 2
√
10
7
(b) Find θ and round to two decimal places.
Answer: 3.58
4.5.10. Find all of the angles θ such that csc(θ) = − √23 . Leave
your answer in exact form.
Answer: θ =
n
4π
3
+ 2πn and θ =
5π
3
+ 2πn for integers,
π
4.5.11. Find an exact value for tan( 12
). It may be helpful to
π/6
π
use the fact that 12 = 2 .
√
√
√
Answer: 2 − 3 = 2+1√3 = 1+3−1
3
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Homework Assignment
Solutions
4.5.12. Find an exact value for
use the fact that 17π
12 =
√
sin( 17π
12 ).
7π
π
+
6
4.
It may be helpful to
Section 4.5
4.5.16. Show that
cot(α + β) =
√ 3
Answer: − 1+
2 2
4.5.13. Show that
cot(α) cot(β) − 1
.
cot(β) + cot(α)
Carefully show each step.
(sin(θ) + cos(θ)) (tan(θ) + cot(θ))
= sec(θ) + csc(θ).
Carefully show each step.
Answer: There are many correct answers.
4.5.17. Show that
cos2
Answer: There are many correct answers.
sec(θ) + 1
θ
=
2
2 sec(2θ)
Carefully show each step.
4.5.14. Show that
2
2
sec (θ) − tan (θ) = 1.
Carefully show each step.
Answer: There are many correct answers.
4.5.15. Show that
1
1
+
= 2 csc2 (θ).
1 − cos(θ) 1 + cos(θ)
Answer: There are many correct answers.
4.5.18. Show that
sin(3x) cos(3x)
−
= 2.
sin(x)
cos(x)
Carefully show each step.
Answer: There are many correct answers.
Carefully show each step.
Answer: There are many correct answers.
Last Updated: March 29, 2014
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