Q: Which of the following is the correct definition of $\tan \theta$ in a right triangle?

$\dfrac{\text{opposite}}{\text{adjacent}}$

Q: If $\vec{u} = \left[ \begin{array}{c} 1 \\ 2 \end{array} \right]$ and $\vec{v} = \left[ \begin{array}{c} 3 \\ 4 \end{array} \right]$, what is $\vec{u} + \vec{v}$?

$\left[ \begin{array}{c} 4 \\ 6 \end{array} \right]$

Q: A vector $\vec{v}$ has magnitude $8$ and direction angle $60^\circ$. What are its components?

$\left[ \begin{array}{c} 8 \cos 60^\circ \\ 8 \sin 60^\circ \end{array} \right]$

Question 1

Given a right triangle with angle $A$, if the adjacent side to $A$ is 3 units and the hypotenuse is 5 units, what is $\cos A$?

Accepted Answer

$\dfrac{3}{5}$

Question 2

Which of the following is the correct definition of $	an 	heta$ in a right triangle?

Accepted Answer

$\dfrac{	ext{opposite}}{	ext{adjacent}}$

Question 3

If $\sin 	heta = \dfrac{1}{2}$ and $	heta$ is in the first quadrant, what is $	heta$?

Accepted Answer

$30^\circ$

Question 4

A triangle has sides of length $a = 2$, $b = 2$, and $c = \sqrt{2}$. What is the measure of angle $C$ opposite side $c$?

Accepted Answer

$45^\circ$

Question 5

Which of the following is the correct value for $\sin 45^\circ$?

Accepted Answer

$\dfrac{\sqrt{2}}{2}$

Question 6

Given the vector $\left[ \begin{array}{c} 3 \ 4 \end{array} ight]$, what is its magnitude?

Accepted Answer

$5$

Question 7

If $\vec{u} = \left[ \begin{array}{c} 1 \ 2 \end{array} ight]$ and $\vec{v} = \left[ \begin{array}{c} 3 \ 4 \end{array} ight]$, what is $\vec{u} + \vec{v}$?

Accepted Answer

$\left[ \begin{array}{c} 4 \ 6 \end{array} ight]$

Question 8

What is the dot product of $\vec{a} = \left[ \begin{array}{c} 2 \ 3 \end{array} ight]$ and $\vec{b} = \left[ \begin{array}{c} 4 \ 1 \end{array} ight]$?

Accepted Answer

$11$

Question 9

If $\vec{v} = \left[ \begin{array}{c} 5 \ 12 \end{array} ight]$, what is the direction angle $	heta$ of $\vec{v}$ with respect to the positive $x$-axis?

Accepted Answer

$\arctan \left( \dfrac{12}{5} \right)$

Question 10

A force of $10$ N acts at an angle of $30^\circ$ above the horizontal. What is the horizontal component of the force?

Accepted Answer

$10 \cos 30^\circ$

Question 11

Which of the following is the correct formula for the magnitude of the vector $\vec{v} = \left[ \begin{array}{c} a \ b \end{array} ight]$?

Accepted Answer

$\sqrt{a^2 + b^2}$

Question 12

If $\vec{u} = \left[ \begin{array}{c} 2 \ -1 \end{array} ight]$ and $\vec{v} = \left[ \begin{array}{c} -1 \ 2 \end{array} ight]$, what is $\vec{u} \cdot \vec{v}$?

Accepted Answer

$-4$

Question 13

A triangle has sides $a = 3$, $b = 4$, and $c = 5$. What is the area of the triangle?

Accepted Answer

$6$

Question 14

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

Accepted Answer

$c^2 = a^2 + b^2 - 2ab \cos C$

Question 15

If $\vec{u} = \left[ \begin{array}{c} 0 \ 1 \end{array} ight]$ and $\vec{v} = \left[ \begin{array}{c} 1 \ 0 \end{array} ight]$, what is the angle between $\vec{u}$ and $\vec{v}$?

Accepted Answer

$90^\circ$

Question 16

A vector $\vec{v}$ has magnitude $8$ and direction angle $60^\circ$. What are its components?

Accepted Answer

$\left[ \begin{array}{c} 8 \cos 60^\circ \ 8 \sin 60^\circ \end{array} ight]$

Question 17

If $\sin 	heta = \dfrac{3}{5}$ and $	heta$ is in the first quadrant, what is $\cos 	heta$?

Accepted Answer

$\dfrac{4}{5}$

Question 18

A vector $\vec{a}$ is perpendicular to $\vec{b}$ if and only if:

Accepted Answer

$\vec{a} \cdot \vec{b} = 0$

Question 19

If $	an 	heta = 1$, what is $	heta$ in degrees in the interval $0^\circ < 	heta < 90^\circ$?

Accepted Answer

$45^\circ$

Fundamentals of Trigonometric Functions and Special Triangles

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