Name:
Math 108 Summer 2015: Exam 1
1 (14pts)
6 (10pts)
2 (17pts)
3 (9pts)
7 (10pts)
EC(5pts)
4 (15pts)
5 (10pts)
Total out of 85
Directions: Put only your answers on the exam. Do all scratch work and side calculation on a
separate piece of scratch paper that is given out in class. Write neatly and legibly. If I can’t read it
you will receive a zero for that problem. Finally be sure to show all of your work (justification) for each
problem to receive full credit.
Deep breaths, trust in yourself. I know you can do this. Good Luck! :)
1
f (θ) = tan(θ)
sin2(θ) + cos2(θ) = 1
SOH-CAH-TOA
2
Conceptual Questions and Short Answer
1. (14 pts total)
(a) (2pts) Choose one of the trig functions f (θ) = sin(θ), g(θ) = cos(θ) or h(θ) = tan(θ) and formally define it
with respect to the unit circle.
(b) (2pts) Choose one of the trig functions f (θ) = sec(θ), g(θ) = csc(θ) or h(θ) = cot(θ) and formally define it
with respect to the unit circle.
(c) (2pts) What is the domain and range of f (θ) = arcsin(θ)?
Domain=
Range=
(d) (2pts) What is the domain and range of f (θ) = arccos(θ)?
Domain=
Range=
(e) (2pts) What is the domain and range of f (θ) = arctan(θ)?
Domain=
Range=
(f) (4pts) Choose one of the trig functions f (θ) = sin(θ), g(θ) = cos(θ) or h(θ) = tan(θ) and explain why we have
to restrict it’s domain in order to define it’s inverse. Use it’s graph in the justification of your response.
3
Computational Questions
2. (17 pts total) For θ = −
4π
3
(a) (5 pts) Plot the angle θ in standard position. Draw in the reference triangle, the reference angle, and the
corresponding sides.
(b) (2 pts) Evaluate sin(θ)
(c) (2 pts) Evaluate cos(θ)
(d) (2 pts) Evaluate tan(θ)
(e) (2 pts) Evaluate sec(θ)
(f) (2 pts) Evaluate csc(θ)
(g) (2 pts) Evaluate cot(θ)
4
3. (9 pts total) For θ =
11π
4
(a) (5 pts) Plot θ on the unit circle. Draw in the reference triangle, the reference angle, and the corresponding sides.
(b) (2 pts) Evaluate tan(θ)
(c) (2 pts) Evaluate sec(θ)
4. (15 pts total) For f (θ) = 2 sin (3θ − π) + 4
(a) (3 pts) Determine the period, P , of f (θ):
(b) (3 pts) Determine the range, R, of f (θ):
P =
R=
(c) (9 pts) Graph one full period of f (θ). Indicate all critical points, e.g. upper and lower bounds, middle symmetry line, 5 critical θs and points.
5
5. (10 pts total) Draw a picture to justify your results for each of the following.
(
)
−1
(a) (5 pts) Evaluate arcsin
.
2
(b) (5 pts) Evaluate arc cot(−1)
(
( (
)))
4π
6. (10 pts) Evaluate tan arccos sin −
. Be sure to show each step in the process to receive full credit.
3
6
7. (10 pts) Solve for all possible values of θ such that 4 sin2 (θ) − 3 = 0. Justify your results by drawing a picture with
reference triangles and angles.
7
8. EXTRA CREDIT (5 pts total)
(a) Given g(θ) = 3 sec(3θ).
i. (1 pt) What is the range of g(θ).
R=
ii. (2 pts) What is the set of all θ such that g(θ) is undefined, i.e. write an expression that represents all of
the vertical asymptotes for the graph of g(θ).
(b) (2 pts) Give a counter example to disprove the claim that arctan (tan(θ)) = θ for every real number θ. Draw
a picture to justify your counter example.
8