Question Convert the given radian measures to degrees and determine the quadrant of the terminal side. a) $\frac{7 \pi}{5}$ degrees, Quad b) $-\frac{11 \pi}{12}$ degrees, Quad

Studdy Solution

STEP 1

1. To convert radians to degrees, we use the conversion factor $180^\circ/\pi$ .
2. Positive angles are measured counterclockwise from the positive x-axis, and negative angles are measured clockwise.
3. The quadrants are determined by the angle's terminal side: - Quadrant I: $0^\circ$ to $90^\circ$ (or $0$ to $\frac{\pi}{2}$ radians) - Quadrant II: $90^\circ$ to $180^\circ$ (or $\frac{\pi}{2}$ to $\pi$ radians) - Quadrant III: $180^\circ$ to $270^\circ$ (or $\pi$ to $\frac{3\pi}{2}$ radians) - Quadrant IV: $270^\circ$ to $360^\circ$ (or $\frac{3\pi}{2}$ to $2\pi$ radians)

STEP 2

1. Convert the radian measure to degree measure.
2. Determine the quadrant where the terminal side lies.
For part a) $\frac{7 \pi}{5}$ :

STEP 3

Convert the radian measure $\frac{7 \pi}{5}$ to degrees.
$\frac{7 \pi}{5} \cdot \frac{180^\circ}{\pi}$

STEP 4

Simplify the expression by canceling $\pi$ and multiplying.
$\frac{7 \cdot 180^\circ}{5}$

STEP 5

Perform the multiplication and division to find the degree measure.
$\frac{7 \cdot 180^\circ}{5} = 7 \cdot 36^\circ = 252^\circ$

STEP 6

Determine the quadrant for $252^\circ$ .
Since $252^\circ$ is greater than $180^\circ$ but less than $270^\circ$ , the terminal side lies in Quadrant III.
For part b) $-\frac{11 \pi}{12}$ :

STEP 7

Convert the radian measure $-\frac{11 \pi}{12}$ to degrees.
$-\frac{11 \pi}{12} \cdot \frac{180^\circ}{\pi}$

STEP 8

Simplify the expression by canceling $\pi$ and multiplying.
$-\frac{11 \cdot 180^\circ}{12}$

STEP 9

Perform the multiplication and division to find the degree measure.
$-\frac{11 \cdot 180^\circ}{12} = -11 \cdot 15^\circ = -165^\circ$

STEP 10

Determine the quadrant for $-165^\circ$ .
Since negative angles are measured clockwise from the positive x-axis, and $-165^\circ$ is between $0^\circ$ and $-180^\circ$ , the terminal side lies in Quadrant II.
The solutions are: a) $252^\circ$ , Quadrant III b) $-165^\circ$ , Quadrant II

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Question Convert the given radian measures to degrees and determine the quadrant of the terminal side. a) $\frac{7 \pi}{5}$ degrees, Quad b) $-\frac{11 \pi}{12}$ degrees, Quad

Studdy Solution

STEP 1

STEP 2

1. Convert the radian measure to degree measure.
2. Determine the quadrant where the terminal side lies.
For part a) $\frac{7 \pi}{5}$ :

STEP 3

Convert the radian measure $\frac{7 \pi}{5}$ to degrees.
$\frac{7 \pi}{5} \cdot \frac{180^\circ}{\pi}$

STEP 4

Simplify the expression by canceling $\pi$ and multiplying.
$\frac{7 \cdot 180^\circ}{5}$

STEP 5

Perform the multiplication and division to find the degree measure.
$\frac{7 \cdot 180^\circ}{5} = 7 \cdot 36^\circ = 252^\circ$

STEP 6

Determine the quadrant for $252^\circ$ .
Since $252^\circ$ is greater than $180^\circ$ but less than $270^\circ$ , the terminal side lies in Quadrant III.
For part b) $-\frac{11 \pi}{12}$ :

STEP 7

Convert the radian measure $-\frac{11 \pi}{12}$ to degrees.
$-\frac{11 \pi}{12} \cdot \frac{180^\circ}{\pi}$

STEP 8

Simplify the expression by canceling $\pi$ and multiplying.
$-\frac{11 \cdot 180^\circ}{12}$

STEP 9

Perform the multiplication and division to find the degree measure.
$-\frac{11 \cdot 180^\circ}{12} = -11 \cdot 15^\circ = -165^\circ$

STEP 10

Determine the quadrant for $-165^\circ$ .
Since negative angles are measured clockwise from the positive x-axis, and $-165^\circ$ is between $0^\circ$ and $-180^\circ$ , the terminal side lies in Quadrant II.
The solutions are: a) $252^\circ$ , Quadrant III b) $-165^\circ$ , Quadrant II

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Question Convert the given radian measures to degrees and determine the quadrant of the terminal side. a) 7π5\frac{7 \pi}{5}57π​ degrees, Quad b) −11π12-\frac{11 \pi}{12}−1211π​ degrees, Quad

Studdy Solution

STEP 1

STEP 2

1. Convert the radian measure to degree measure.2. Determine the quadrant where the terminal side lies.For part a) 7π5\frac{7 \pi}{5}57π​:

STEP 3

Convert the radian measure 7π5\frac{7 \pi}{5}57π​ to degrees.7π5⋅180∘π \frac{7 \pi}{5} \cdot \frac{180^\circ}{\pi} 57π​⋅π180∘​

STEP 4

Simplify the expression by canceling π\piπ and multiplying.7⋅180∘5 \frac{7 \cdot 180^\circ}{5} 57⋅180∘​

STEP 5

Perform the multiplication and division to find the degree measure.7⋅180∘5=7⋅36∘=252∘ \frac{7 \cdot 180^\circ}{5} = 7 \cdot 36^\circ = 252^\circ 57⋅180∘​=7⋅36∘=252∘

STEP 6

Determine the quadrant for 252∘252^\circ252∘.Since 252∘252^\circ252∘ is greater than 180∘180^\circ180∘ but less than 270∘270^\circ270∘, the terminal side lies in Quadrant III.For part b) −11π12-\frac{11 \pi}{12}−1211π​:

STEP 7

Convert the radian measure −11π12-\frac{11 \pi}{12}−1211π​ to degrees.−11π12⋅180∘π -\frac{11 \pi}{12} \cdot \frac{180^\circ}{\pi} −1211π​⋅π180∘​

STEP 8

Simplify the expression by canceling π\piπ and multiplying.−11⋅180∘12 -\frac{11 \cdot 180^\circ}{12} −1211⋅180∘​

STEP 9

Perform the multiplication and division to find the degree measure.−11⋅180∘12=−11⋅15∘=−165∘ -\frac{11 \cdot 180^\circ}{12} = -11 \cdot 15^\circ = -165^\circ −1211⋅180∘​=−11⋅15∘=−165∘

STEP 10

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Question Convert the given radian measures to degrees and determine the quadrant of the terminal side. a) $\frac{7 \pi}{5}$ degrees, Quad b) $-\frac{11 \pi}{12}$ degrees, Quad

1. Convert the radian measure to degree measure.
2. Determine the quadrant where the terminal side lies.
For part a) $\frac{7 \pi}{5}$ :

Convert the radian measure $\frac{7 \pi}{5}$ to degrees.
$\frac{7 \pi}{5} \cdot \frac{180^\circ}{\pi}$

Simplify the expression by canceling $\pi$ and multiplying.
$\frac{7 \cdot 180^\circ}{5}$

Perform the multiplication and division to find the degree measure.
$\frac{7 \cdot 180^\circ}{5} = 7 \cdot 36^\circ = 252^\circ$

Determine the quadrant for $252^\circ$ .
Since $252^\circ$ is greater than $180^\circ$ but less than $270^\circ$ , the terminal side lies in Quadrant III.
For part b) $-\frac{11 \pi}{12}$ :

Convert the radian measure $-\frac{11 \pi}{12}$ to degrees.
$-\frac{11 \pi}{12} \cdot \frac{180^\circ}{\pi}$

Simplify the expression by canceling $\pi$ and multiplying.
$-\frac{11 \cdot 180^\circ}{12}$

Perform the multiplication and division to find the degree measure.
$-\frac{11 \cdot 180^\circ}{12} = -11 \cdot 15^\circ = -165^\circ$