# Degree Equivalent of π

PART A – MULTIPLE CHOICE (10 MARKS)

1. (1 mark) When convert radians, 60◦ will be equivalent to

A. π

2

B. 2π

3

C. π

3

D. π

4

2. (1 mark) What the degree equivalent of π

6 radians?

A. 45◦

B. 60◦

C. 72◦

D. 90◦

3. (1 mark) An arc subtends an angle of 2 radians in the centre of a circle of radius 8 cm. What is the length of this arc?

A. 4 cm

Eton Academy 1 North York, Canada

MHF4U: Test – 4 Revision – January 15, 2022

B. 6 cm

C. 12 cm

D. 16 cm

4. (1 mark) Considering the equation sin x = cos x, which of the following is NOT true?

A. it is an identity

B. x = π

4 satisfies the equation

C. x = 0 does not satisfy the equation

D. x = 9π

4 satisfies the equation

5. (1 mark) What is the maximum value of the transformed sine function g(x) =

−2 sin [ 3 ( x +

π

4

)] − 6?

A. −4

B. −8

C. 4

D. 3

6. (1 mark) The double-angle expansion for cos 2x is

A. cos2 x− 2 sin x

B. cos2 x− sin2 x

C. sin2 x− cos2 x

D. sin2 x− 2 cos x

7. (1 mark) If the interval under consideration is 0 ≤ x ≤ 2π, how many solutions does the equation sin(x)(1 − cos x) = 0 have?

A. 1

B. 2

C. 3

D. 4

8. (1 mark) Within the interval 0 ≤ x ≤ 2π, the graph of sin x crosses the x-axis at the three points: x = 0,π, 2π. After a transformation of the form sin kx, with k ≥ 1.5, how are the x-intercepts affected within the same interval?

A. the new graph has three x-intercepts as before.

Eton Academy 2 North York, Canada

MHF4U: Test – 4 Revision – January 15, 2022

B. the new graph has more than three x-intercepts.

C. the new graph has less than three x-intercepts.

D. the new graph does not have any x-intercepts.

9. (1 mark) The equation sin(x) csc(x) = 1…

A. is an identity.

B. has no solution.

C. is true for only positive values of x.

D. is true for only negative values of x.

10. (1 mark) The value of tan

( −

3π

2

) is

A. 0

B. 1

C. −1

D. ∞

PART B – OTHER QUESTIONS (90 MARKS)

11. (10 marks) Let α and β be acute angles in quadrant I, with sin α = 7

25 and

cos β = 5

13 . Without using a calculator, determine the exact values of sin(α + β) and

tan(α + β).

12. (10 marks) The quadratic trigonometric equation

cot2(x) − b cot(x) + c = 0

has the solutions π

6 , π

4 ,

7π

6 , and

5π

4 in the interval 0 ≤ x ≤ 2π. What are the values of

b and c?

13. (10 marks) Suppose that

2 cos2(x) + 4 sin(x) cos(x) = a sin(2x) + b cos(2x) + c

is an identity, determine the values of a, b and c.

14. (10 marks) Let x = tan A and y = tan 2A.

a) Eliminate A between the two trigonometric functions to express y in terms of x.

b) What type of function was obtained in a) above?

Eton Academy 3 North York, Canada

MHF4U: Test – 4 Revision – January 15, 2022

15. (10 marks) Each person’s blood pressure is different, but there is a range of blood pressure values that is considered healthy. The function

p(t) = −20 cos (

5π

3 t

) + 100

models the blood pressure, p, in millimetres of mercury, at time t, in seconds, of a person at rest.

a) What is the period of the function? What does the period represent for an individ- ual?

b) How many time does this person’s heart beat each minute?

c) Sketch the graph of p(t) within the interval 0 ≤ t ≤ 6.

d) What is the range of the function? Explain the meaning of the range in terms of a person’s blood pressure.

16. (10 marks) A rung on a hamster wheel, with a radius of 25 cm, is travelling at a constant speed. It makes one complete revolution in 3 s. The axle of the hamster wheel is 27 cm above the ground.

a) Sketch a graph of the height of the rung above the ground during two complete revolutions, beginning when the rung is closet to the ground.

b) Describe the transformations necessary to convert y = cos(x) into the function you graphed in part a)

c) Write the equation that models this situation.

17. (10 marks) Create an equivalent expression for cos

( x +

3π

2

) , using an appropriate

compound angle formula.

18. (10 marks) You are asked to evaluate tan 135◦ by using either the compound angle expression of tan(45◦ + 90◦), or the expansion of tan(60◦ + 75◦). With justification, which of these two expansions will you choose?

19. (10 marks) Produce a counter-example to show that

a) cos(2x) 6= 2 cos(x)

b) sin(x + y) 6= sin(x) + sin(y)