Q: Which of the following is the correct definition of the sine function for an angle $$\theta in a $$right triangle?

$$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$$

Q: A Ferris wheel has a radius of 20 meters. If a passenger travels through an angle of $$\frac{\pi}{3}$$ radians, what is the length of the arc they travel?

$$\frac{20\pi}{3}$$ \text{ meters}$

Q: A point moves in a circle of radius 5 cm at a constant speed, making one revolution every 4 seconds. What is its linear speed?

$$\frac{5\pi}{2}$$ \text{ cm/s}$

Question 1

Which of the following is the correct definition of the sine function for an angle $$	heta in a $$right triangle?

Accepted Answer

$$\sin 	heta = \frac{	ext{opposite}}{	ext{hypotenuse}}$$

Question 2

Convert $$120^\circ to $$radians.

Accepted Answer

$$\frac{2\pi}{3}$$

Question 3

A Ferris wheel has a radius of 20 meters. If a passenger travels through an angle of $$\frac{\pi}{3}$$ radians, what is the length of the arc they travel?

Accepted Answer

$$\frac{20\pi}{3}$$ 	ext{ meters}$

Question 4

Which of the following is an identity for $$\sin^2$ 	heta + \cos^2$ 	heta$$?

Accepted Answer

$$1$$

Question 5

If $$	an 	heta = 3$$ and $$	heta is in $$the first quadrant, what is $$\sin 	heta$$?

Accepted Answer

Question 6

Which of the following is the correct double-angle identity for $$\sin 2	heta$$?

Accepted Answer

$$2 \sin 	heta \cos 	heta$$

Question 7

Solve for $$x in $$the interval $$[0, 2\pi)$$: $$2\sin x - 1 = 0.$$

Accepted Answer

$$x = \frac{\pi}{6}, \frac{5\pi}{6}$$

Question 8

A ship sails 10 km on a bearing of $$30^\circ$$ and then 20 km on a bearing of $$120^\circ. $$What is the distance between the starting and ending points? (Use the Law of Cosines.)

Accepted Answer

$$\sqrt{10^2$ + 20^2$ - 2 \cdot 10 \cdot 20 \cdot \cos 90^\circ}$$

Question 9

Which of the following is the correct expression for the dot product of vectors $$\vec{a} = \langle 2, 3 \rangle$$ and $$\vec{b} = \langle 4, -1 \rangle$$?

Accepted Answer

$$2 \cdot 4 + 3 \cdot (-1) = 5$$

Question 10

Express the complex number $$z = 1 + i\sqrt{3} in $$trigonometric (polar) form.

Accepted Answer

$$2 \left( \cos \frac{\pi}{3} + i \sin \frac{\pi}{3} \right)$$

Question 11

Which of the following is the correct sum-to-product identity for $$\sin A + \sin B$$?

Accepted Answer

$$2 \sin \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right)$$

Question 12

A ramp rises 3 meters over a horizontal distance of 12 meters. What is the angle of elevation of the ramp, to the nearest degree?

Accepted Answer

$$14^\circ$$

Question 13

If $$\cos 	heta = -\frac{1}{2}$$ and $$	heta is in $$the third quadrant, what is $$	heta in $$radians?

Accepted Answer

$$\frac{4\pi}{3}$$

Question 14

Which of the following is the correct formula for the area of a triangle given two sides $$a$$ and $$b$$ and the included angle $$C$$?

Accepted Answer

$$\frac{1}{2}ab\sin C$$

Question 15

Given $$	an$$^{-1}$$(1)$$, what is the principal value in radians?

Accepted Answer

$$\frac{\pi}{4}$$

Question 16

Which of the following is the correct half-angle identity for $$\sin^2$ \frac{	heta}{2}$$?

Accepted Answer

$$\frac{1 - \cos 	heta}{2}$$

Question 17

A point moves in a circle of radius 5 cm at a constant speed, making one revolution every 4 seconds. What is its linear speed?

Accepted Answer

$$\frac{5\pi}{2}$$ 	ext{ cm/s}$

Question 18

Which of the following is the correct polar form of the complex number $$-1 + i\sqrt{3}$$?

Accepted Answer

$$2 \left( \cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3} \right)$$

Trigonometry Syllabus and Course Structure – MATH 1316

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