MATH1060: Midterm 3 Practice Problems
The following are practice problems for the first exam.
1. Find all solutions to the following trigonometric equations:
π
2π
+ nπ,
+ nπ
3
3
π
(b) csc ν + cot ν = 1 ν = + 2πn
2
π
nπ
(c) tan(3η) − 1 = 0 η =
+
12
3
(d) This question was impossible to solve analytically.
(a) 4 cos2 φ − 1 = 0 φ =
2. Find the exact value of each of the following expressions:
√
√
6− 2
(a) sin(π/12)=
4
√
3
−
3
(b) tan(165◦ )= √
3+3
√
3
◦
◦
◦
◦
(c) cos(18 ) cos(12 ) − sin(18 ) sin(12 )=
2
√
π
π
π
π
3
(d) sin
cos + sin cos =
12
4
4
12
2
√
6
(e) cos 75◦ + cos 15◦ =
2
3. Find the exact solutions to the following trigonometric equations in the interval [0, 2π)
(a) cos 2χ − cos χ = 0. χ = 0,
2π 4π
,
3 3
(b) tan 2ν − 2 cos ν = 0. This one is tough. ν =
(c) 4 − 8 sin2 µ = 0. µ =
π 3π 5π
,
,
6 2 6
π 3π 5π 7π
,
,
,
4 4 4 4
ρ
(d) sin + cos ρ = 0. ρ = π. There is a subtlety in this problem (and problems like it) that
2
we did not discuss in class. This subtlety will not come up on the exam, however there
may be a problem similar to this one. Briefly, the issue is that since the original equation
ρ
has the angle , if we solve for ρ, and add an integer multiple of 2π to get a coterminal
2
ρ + 2πn
ρ
angle, then
will not necessarily be coterminal to . In class, we solved this
2
2
−π
5π
problem and determined
is a solution. Then we hastily concluded that
is also a
3
3
5π
solution. A quick check shows that
is not a solution, so the only solution in [0, 2π)
3
is ρ = π. Again, this will not be an issue you need to worry about on the exam. It is
easily dealt with, though.
1
(e) sin
β
π 5π
+ cos β = 1. β = 0, ,
2
3 3
4. Use the power reducing formula to rewrite the expression sin4 x cos2 x in terms of the first
1
[1 − cos(2x) − cos(4x) − cos(2x) cos(4x)]
power of cosine.
16
5. Use the sum-to-product or product-to-sum formula to rewrite each expression:
1
[cos(2γ) − cos(8γ)]
2
(b) cos 6δ + cos 2δ= 2 cos(4δ) cos(2δ)
r
(a) sin 5γ sin 3γ=
6. Use the half-angle formula to simplify
7. If sin u =
1 − cos 14x
. sin(7x)
2
p
7
and u lies in the second quadrant, find cos(u/2). 1/50
25
8. Use any means you like to solve the triangle with α = 24.3◦ , γ = 54.6◦ , and c = 10.3.
β = 101.1◦ , a = 5.20, b = 12.40
9. Use any means you like to solve the triangle with α = 120◦ , β = 45◦ , and c = 16. γ = 15◦ ,
a = 53.54, b = 43.71
10. Use any means you like to solve the triangle with β = 63.2◦ , γ = 47.6◦ and b = 12.2.
α = 69.2◦ , a = 12.78, c = 10.09
11. Use any means you like to solve the triangle with α = 110◦ , a = 125, and b = 100. β1 = 48.74◦ ,
γ1 = 21.26◦ , c1 = 48.23. Only one solution.
12. Use any means you like to solve the triangle with β = 100◦ , b = 14, and c = 19. Impossible.
13. Use any means you like to solve the triangle with α = 28◦ , b = 12.8, and a = 8. β1 = 48.69◦ ,
γ1 = 103.31◦ , c1 = 16.58, and β2 = 131.31◦ , γ2 = 20.69◦ , c2 = 6.42
√
14. Use Heron’s Formula to find the area of a triangle with side lengths 7, 8, and 9. Answer: 12 5
15. Use any means you like to solve the triangle with α = 50◦ , b = 15 and c = 30. a = 23.38,
β = 29.44◦ , γ = 100.56◦
16. Use any means you like to solve the triangle with side lengths 7, 8, and 9. β = 48.19◦ ,
α = 73.40◦ , γ = 58.41◦ . Depending on how you labelled the angles, your answer may differ,
but the numbers should be the same.
17. Write the vector, ~v , with initial point (−4, 5) and terminal point (3, −1) in standard form.
Answer: ~v = h7, −6i
18. Write the vector from the last question as a linear combination of the standard unit vectors
ı̂ and ̂. Answer: ~v = 7ı̂ − 6̂
√
19. Compute the magnitude of the vector, ~v , from the last question. Answer: k~v k = 85
20. Let ~v = h3, −1i and w
~ = h−1, 4i. Write 3~v − 2w
~ in standard form. Answer: h11, −11i
2