Pre-Calculus Math 30
Trigonometry
Mid-Unit Quiz
S12021
Version A
Name: ____________________________
Provide a clear, complete and organized algebraic solution for each question. Presentation and notation are
important. Show all your work to receive full marks.
Use the following information to answer question 1.
An angle, 𝜃, in standard position is shown below.
(1.) The best estimate of the rotation angle, 𝜃, is
A. 1.25 radians
B. 3.12 radians
C. 4.01 radians
[1]
D. 5.38 radians
(2.) The terminal arm of an angle, 𝜃, when drawn in standard position, contains the point 𝑃(𝑥, 𝑦) where P is
7
on the unit circle. If sin 𝜃 = and tan 𝜃 < 0, then what is the exact value of cos 𝜃?
[2]
10
Pre-Calculus Math 30
Trigonometry
Mid-Unit Quiz
S12021
Version A
Use the following information to answer questions 3 and 4.
A circle with a radius, r, an indicated arc length 34𝜋, and two central angles of
𝑎
𝜋
15
and
17
𝜋
15
is shown below.
15
1
15
34
(3.) For the angle
𝑎
𝜋,
15
the value of a is ____.
(4.) The length of the radius, r, of the circle, to the nearest whole number, is _____.
[1]
[1]
(5.) The range of 𝑦 = asin(𝑥) + 𝑏 is −1 ≤ 𝑦 ≤ 5. If a is a positive, then the values of a and b are, respectively,
_______ and _______.
12
[2]
1
(6.) Given cos 𝜃 = − 13 and −𝜋 ≤ 𝜃 ≤ − 2 𝜋. Determine the exact values of csc 𝜃 and tan 𝜃.
[3]
Pre-Calculus Math 30
Trigonometry
Mid-Unit Quiz
S12021
Version A
Use the following information to answer question 7.
For the graph of the function 𝑓(𝑥) = −3 sin[2(𝑥 − 5)] + 𝑑, the following statements were made.





Statement 1: The amplitude is 3.
Statement 2: The maximum value is (𝑑 − 3).
Statement 3: The period is 2𝜋.
Statement 4: If 𝑑 > 3, then the graph of 𝑦 = 𝑓(𝑥) will have no x-intercepts.
Statement 5: When compared to the graph of 𝑔(𝑥) = −3 sin(2𝑥) + 𝑑, the graph of 𝑦 = 𝑓(𝑥)
has been horizontally translated 5 units right.
(7.) The true statements are
[1]
A. 1, 5 only
B. 2, 3 only
C. 1, 4, 5 only
D. 2, 4, 5 only
Use the following information to answer question 8.
Each of the trigonometric ratios listed below results in a value of zero or undefined.
1
tan ( 𝜋)
2
3
cot ( 𝜋)
2
sin(𝜋)
csc(2𝜋)
(8.) Using the following code to indicate that the value of the ratio is zero or that the ratio is undefined. [2]
1= The value of the ratio is zero
2= The value of the ratio is undefined.
Ratio:
___________
1
tan ( 𝜋)
2
__________
3
cot ( 𝜋)
2
__________
sin(𝜋)
__________
csc(2𝜋)
Pre-Calculus Math 30
Trigonometry
Mid-Unit Quiz
S12021
Version A
Use the following information to answer question 9.
𝜋
8
The partial graph of the sine function below has a minimum point at ( , 1) and a maximum
5𝜋
point at ( 8 , 11) as shown below. The equation of the function can be expressed in the
form 𝑦 = asin(𝑏(𝑥 − 𝑐)) + 𝑑 where 𝑎, 𝑏, 𝑐, 𝑑 ∈ 𝑊.
(9.) With a minimum phase shift, the values of a, b, c, and d are, respectively, _______, _______, _______,
and _______.
[2]
(10.) Given the following graph, determine the simplest equation in terms of cosine.
[2]
Pre-Calculus Math 30
Trigonometry
Mid-Unit Quiz
S12021
Version A
(11.) A paddle wheel with a diameter of 40 metres has its centre 8 metres above the surface of the water. The
wheel rotates once every 30 seconds. If the bucket starts at the lowest point …
A. Determine the equation that would describe the buckets’ height above water as
a function of time.
[2]
B. Find the height of the bucket 12 seconds after the wheel starts to rotate, to the
nearest tenth.
[1]
C. How long is the bucket above the surface of the water in one rotation of the wheel? Nearest hundredth
[1]
D. Determine the angular speed of the paddle wheel, as an exact value in radians/minute.
3
(12.) Given the point 𝑃(𝜃) = (− 5 ,
4
).
5
1
Determine the coordinates of 𝑃 (𝜃 + 2 𝜋).
[1]
[1]
(13.) Given 𝑓(𝜃) = 𝑐𝑜𝑠(𝑛𝜃) has the same period as the graph of 𝑔(𝜃) = tan 𝜃, the value of n is _____. [1]
Pre-Calculus Math 30
Trigonometry
Mid-Unit Quiz
S12021
Version A
(14.) Determine the exact value of each of the following:
17
a. csc(−240°)=
b. cot ( 6 𝜋) =
[2 each]
c. 𝑠𝑒𝑐 (−
11
𝜋)
4
=
(15.) Solve each of the following over the stated domain. The answers must be given in the units designated
by the domain.
1
a. cos 𝜃 = − 2 ; 0° ≤ 𝜃 ≤ 360°
[2]
b. 2 sin 𝜃 − √3 = 0 ; 0 ≤ 𝜃 ≤ 2𝜋
[2]
c. tan 𝜃 = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 ; 0 ≤ 𝜃 ≤ 2𝜋
[2]
Use the following information to answer question 16.
The point (2𝑎, −𝑎), where 𝑎 > 0 is the point of intersection between the terminal arm of an angle, θ, in
𝑚
standard position and the unit circle. The value of a can be written in the form , where m and n are
√𝑛
single digit whole numbers.
(16.) The values of m and n, respectively are ______ and ______.
[1]