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 $$0^\circ \u003c 	heta \u003c 90^\circ$$?

Accepted Answer

$$45^\circ$$

Fundamentals of Trigonometric Functions and Special Triangles

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