Q: A Ferris wheel has a radius of 20 meters. What is the length of the arc subtended by a central angle of $$\frac{\pi}{3}$$ radians?

$$20 \times \frac{\pi}{3} = \frac{20\pi}{3}$$ meters

Q: A surveyor measures the angle of elevation to the top of a building as $$60^\circ$$ from a point 30 meters away. What is the height of the building?

$$30 \times \tan 60^\circ = 30 \times \sqrt{3} \approx 51.96$$ meters

Q: Which of the following is the correct formula for converting degrees to radians?

$$\text{radians} = \text{degrees} \times \frac{\pi}{180}$$

Q: A vector $$\vec{u}$$ has components $$\langle -6, 8 \rangle. $$What is the direction angle $$\theta of$$ $$\vec{u}$$ measured counterclockwise from the positive $$x-$$axis?

$$\theta = \arctan\left(\frac{8}{-6}\right) + 180^\circ \approx 126.87^\circ$$

Q: Which of the following is the correct statement of the Pythagorean Identity?

$$\sin^2$ \theta + \cos^2$ \theta = 1$$

Question 1

Which of the following is the correct definition of the radian measure of an angle?

Accepted Answer

The ratio of the length of the arc to the radius of the circle.

Question 2

A Ferris wheel has a radius of 20 meters. What is the length of the arc subtended by a central angle of $$\frac{\pi}{3}$$ radians?

Accepted Answer

$$20 	imes \frac{\pi}{3} = \frac{20\pi}{3}$$ meters

Question 3

Which trigonometric function is defined as the ratio of the adjacent side to the hypotenuse in a right triangle?

Accepted Answer

$$\cos 	heta$$

Question 4

If $$\sin 	heta = \frac{3}{5}$$ and $$	heta is in $$Quadrant II, what is $$\cos 	heta$$?

Accepted Answer

$$-\frac{4}{5}$$

Question 5

A surveyor measures the angle of elevation to the top of a building as $$60^\circ$$ from a point 30 meters away. What is the height of the building?

Accepted Answer

$$30 	imes 	an 60^\circ = 30 	imes \sqrt{3} \approx 51.96$$ meters

Question 6

Which of the following is the correct formula for the Law of Sines?

Accepted Answer

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

Question 7

Given $$\vec{v} = \langle 3, 4 \rangle$$, what is the magnitude of $$\vec{v}$$?

Accepted Answer

$$\sqrt{3^2$ + 4^2$} = 5$$

Question 8

A plane flies 500 miles east, then 300 miles north. What is the straight-line distance from its starting point?

Accepted Answer

$$\sqrt{500^2$ + 300^2$} = \sqrt{340000} \approx 583.1$$ miles

Question 9

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

Accepted Answer

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

Question 10

If $$	an 	heta = 2$$ and $$	heta is in $$Quadrant III, what is $$\sin 	heta$$?

Accepted Answer

$$-\frac{2}{\sqrt{5}}$$

Question 11

Which of the following is the correct formula for converting degrees to radians?

Accepted Answer

$$	ext{radians} = 	ext{degrees} 	imes \frac{\pi}{180}$$

Question 12

A vector $$\vec{u}$$ has components $$\langle -6, 8 angle. $$What is the direction angle $$	heta of$$ $$\vec{u}$$ measured counterclockwise from the positive $$x-$$axis?

Accepted Answer

$$	heta = \arctan\left(\frac{8}{-6}ight) + 180^\circ \approx 126.87^\circ$$

Question 13

Which of the following is the correct statement of the Pythagorean Identity?

Accepted Answer

$$\sin^2$ 	heta + \cos^2$ 	heta = 1$$

Question 14

A ship sails 100 km at a bearing of $$30^\circ$$ east of north. What are its displacement components?

Accepted Answer

$$x = 100 \sin 30^\circ = 50 km$$, $$y = 100 \cos 30^\circ \approx 86.6 km$$

Question 15

Which of the following is the correct formula for the Law of Cosines?

Accepted Answer

$$a^2$$ = $$b^2$$ + $$c^2 - 2bc$$ \cos A$$

Question 16

If $$\sin \alpha = 0.6$$ and $$\cos \alpha = 0.8$$, what is $$	an \alpha$$?

Accepted Answer

$$\frac{0.6}{0.8} = 0.75$$

Question 17

A triangle has sides $$a = 7$$, $$b = 10$$, and angle $$C = 45^\circ. $$What is the area of the triangle?

Accepted Answer

$$\frac{1}{2} 	imes 7 	imes 10 	imes \sin 45^\circ = 35 	imes \frac{\sqrt{2}}{2} \approx 24.75$$

Question 18

Which of the following is the correct formula for the dot product of two vectors $$\vec{a} = \langle a_1$, a_2$ \rangle$$ and $$\vec{b} = \langle b_1$, b_2$ \rangle$$?

Accepted Answer

$$\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2$$

Question 19

A complex number $$z = 4 + 3i is $$represented in polar form as $$r(\cos 	heta + i \sin 	heta). $$What are $$r$$ and $$	heta$$?

Accepted Answer

$$r = 5$$, $$	heta = \arctan\left(\frac{3}{4}ight) \approx 36.87^\circ$$

MAC 1114: Trigonometry Syllabus and Course Overview

Guida di studio - Domande di pratica

Study guide - Flashcard