MATH1060: Midterm 2 Practice Problems
The following are practice problems for the first exam.
1. Evaluate the following expressions:
√
3 π
=
(a) arcsin
2
3
π
−1
(b) cos (0) =
2
−1
−π
(c) sin−1
=
2
6
√
π
(d) arctan 3 =
3
√
− 2 3π
(e) arccos
=
2
4
−π
(f) tan−1 (−1) =
4
2. Sketch the graph of the following functions:
x
(a) y = tan
2
(b) y = csc(x)
(c) y = arctan(x)
(d) y = sin−1 (x)
Use a calculator
3. The sun is 20◦ above the horizon. If Mike is 6 feet tall, how long is his shadow? Answer:
6
s=
tan 20◦
4. An airplane is 200 miles north and 125 miles west of the airport. The pilot wants to fly directly
125
to the airport. What bearing should be taken? Answer: SθE where θ = tan−1
200
5. Determine the angle between the diagonal √
of a cube and its edge. (This is Exercise 44 from
§4.8 in the textbook). Answer: θ = tan−1 ( 2)
6. A car is moving at 60 miles per hour. Its wheel is rotating at 2 revolutions per second. What
is the radius of its tire? Answer: 7.002 feet
7. A satellite in a circular orbit 1250 km above earth makes one complete revolution every 110
minutes. If we assume that Earth is a sphere of radius 6378 km, what is the linear speed of
the satellite? Answer: 430 kilometers per minute
8. Identify each of the following expressions as one of the standard trigonometric functions:
(a) sec β csc β − cot β Answer: tan β
1
(b)
sin γ
Answer: csc γ
1 − cos2 γ
9. Simplify the following expressions:
(a) sec4 α − tan4 α Answer: 1 + 2 tan2 α
(b) 1 − 2 cos2 δ + cos4 δ Answer: sin4 δ
(c) sin2 ζ + 3 cos ζ + 3 Answer: −(cos ζ − 4)(cos ζ + 1)
(d) sin tan + cos Answer: sec (e) sin η(csc η − sin η) Answer: cos2 η
(f) tan ι −
sec2 ι
Answer: − cot ι
tan ι
10. Rewrite the following expression so it is not in fractional form:
6
tan κ + sec κ
Answer: −6(tan κ − sec κ
11. Use the trigonometric substitution 3x = 5 tan λ to simplify the expression
5 sec λ
12. Simplify the expression ln | cos µ| + ln(1 + tan2 µ)
13. Verify the following identities:
(a) cos2 α − sin2 α = 1 − 2 sin2 α
1
1
(b)
+
= tan γ + cot γ
tan γ
cot γ
(c) cos2 θ + cos2 ( π2 − θ) = 1
x
(d) tan(sin−1 x) = √
1 − x2
2
p
9x2 + 25 Answer: