Question

question 4
a central angle in a circle has a measure of 5 radians with an intercepted arc length of 50 inches.
what is the length of the radius of the circle?
type your answer in the box provided. only type in numbers - do not include the word inches.

question 5
an angle in standard position has a measure of - 473°. find a coterminal angle in degrees that has a measure between 0° and 360°.
type in your answer to the box below. only type in a number - do not type in the word degrees.

question 6
which angle in standard position would terminate in quadrant 3?
-\frac{6\pi}{7}
-\frac{4\pi}{9}
\frac{7\pi}{8}
\frac{16\pi}{9}

Explanation:

Question 4

Step1: Recall arc - length formula

The formula for arc - length $s$ is $s = r\theta$, where $s$ is the arc - length, $r$ is the radius, and $\theta$ is the central angle in radians. We need to solve for $r$, so $r=\frac{s}{\theta}$.

Step2: Substitute given values

Given $s = 50$ inches and $\theta=5$ radians. Substituting into the formula $r=\frac{50}{5}$.
$r = 10$

Question 5

Step1: Use coterminal - angle formula

Coterminal angles are given by $\alpha=\beta + 360n$, where $\alpha$ and $\beta$ are angles and $n$ is an integer. We want to find $n$ such that $0<\beta + 360n<360$ for $\beta=-473^{\circ}$.
Let's start by adding $360^{\circ}$ multiple times to $- 473^{\circ}$.
First, add $360^{\circ}$ once: $-473 + 360=-113^{\circ}$.
Add $360^{\circ}$ again: $-113+360 = 247^{\circ}$

Question 6

Step1: Determine range of angles in Quadrant 3

The range of angles in standard position that terminate in Quadrant 3 is $\pi+2k\pi<\theta<\frac{3\pi}{2}+2k\pi$, $k\in\mathbb{Z}$.
For $k = 0$, the range is $\pi<\theta<\frac{3\pi}{2}\approx3.14 <\theta<4.71$.

• For $\theta=-\frac{6\pi}{7}\approx - 2.69$, it is in Quadrant 2.
• For $\theta=-\frac{4\pi}{9}\approx - 1.40$, it is in Quadrant 4.
• For $\theta=\frac{7\pi}{8}\approx2.75$, it is in Quadrant 2.
• For $\theta=\frac{16\pi}{9}\approx5.58$. Since $\frac{16\pi}{9}= \pi+\frac{7\pi}{9}$ and $\pi<\frac{16\pi}{9}<\frac{3\pi}{2}$, it is in Quadrant 3.

Answer:

Question 4: 10
Question 5: 247
Question 6: $\frac{16\pi}{9}$