Question

name: student number instructor course section math 1012/1112 lab 38 trigonometry: basic concepts for all questions, work must be shown. write your final solution in the space provided to complete the questions in this lab, use your class notes, homework examples and your text /course manual as resources. 1. convert each radian measure to degree measure. a) \\( \frac { 5 \pi } { 12 } \\) 1a) b) \\( - \frac { 11 \pi } { 6 } \\) 1b) 2. convert each degree measure to radian measure. a) \\( - 225 ^ { \circ } \\) 2a) b) \\( 990 ^ { \circ } \\) 2b) 3. identify the quadrant in which the terminal ray of the given angle lies and state the reference angle. a) \\( \frac { 3 \pi } { 4 } \\) quadrant reference angle b) \\( - \frac { 38 \pi } { 3 } \\) quadrant reference angle

Explanation:

1a) Convert $\frac{5\pi}{12}$ radians to degrees

Step1: Use the conversion formula

To convert radians to degrees, use the formula $D = R\times\frac{180^{\circ}}{\pi}$, where $D$ is the degree measure and $R$ is the radian measure.
For $R=\frac{5\pi}{12}$, we have $D=\frac{5\pi}{12}\times\frac{180^{\circ}}{\pi}$.

Step2: Simplify the expression

Cancel out the $\pi$ terms: $\frac{5}{12}\times180^{\circ}$.
Calculate $\frac{5\times180^{\circ}}{12}=\frac{900^{\circ}}{12} = 75^{\circ}$.

1b) Convert $-\frac{11\pi}{6}$ radians to degrees

Step1: Apply the conversion formula

Using $D = R\times\frac{180^{\circ}}{\pi}$ with $R =-\frac{11\pi}{6}$, we get $D=-\frac{11\pi}{6}\times\frac{180^{\circ}}{\pi}$.

Step2: Simplify

Cancel $\pi$: $-\frac{11}{6}\times180^{\circ}$.
Calculate $-\frac{11\times180^{\circ}}{6}=- 330^{\circ}$.

2a) Convert $-225^{\circ}$ to radians

Step1: Use the conversion formula

The formula to convert degrees to radians is $R = D\times\frac{\pi}{180^{\circ}}$, where $D=-225^{\circ}$.
So, $R=-225^{\circ}\times\frac{\pi}{180^{\circ}}$.

Step2: Simplify

Simplify the fraction: $\frac{-225}{180}\pi=\frac{-5}{4}\pi=-\frac{5\pi}{4}$.

2b) Convert $990^{\circ}$ to radians

Step1: Apply the conversion formula

Using $R = D\times\frac{\pi}{180^{\circ}}$ with $D = 990^{\circ}$, we have $R=990^{\circ}\times\frac{\pi}{180^{\circ}}$.

Step2: Simplify

Simplify $\frac{990}{180}\pi=\frac{11}{2}\pi$.

3a) Analyze $\frac{3\pi}{4}$

Step1: Determine the quadrant

Since $\frac{\pi}{2}<\frac{3\pi}{4}<\pi$, the terminal ray of $\frac{3\pi}{4}$ lies in the second quadrant.

Step2: Calculate the reference angle

The reference angle formula for an angle $\theta$ in the second quadrant is $\pi-\theta$.
For $\theta=\frac{3\pi}{4}$, the reference angle is $\pi-\frac{3\pi}{4}=\frac{\pi}{4}$.

3b) Analyze $-\frac{38\pi}{3}$

Step1: Find a coterminal positive angle

Add $2\pi n$ (where $n$ is an integer) to make the angle positive.
$-\frac{38\pi}{3}+14\pi=-\frac{38\pi}{3}+\frac{42\pi}{3}=\frac{4\pi}{3}$.

Step2: Determine the quadrant

Since $\pi<\frac{4\pi}{3}<\frac{3\pi}{2}$, the terminal ray lies in the third quadrant.

Step3: Calculate the reference angle

The reference angle formula for an angle $\theta$ in the third quadrant is $\theta - \pi$.
For $\theta=\frac{4\pi}{3}$, the reference angle is $\frac{4\pi}{3}-\pi=\frac{\pi}{3}$.

Answer:

1a) $75^{\circ}$
1b) $-330^{\circ}$
2a) $-\frac{5\pi}{4}$
2b) $\frac{11\pi}{2}$
3a) quadrant: second, reference angle: $\frac{\pi}{4}$
3b) quadrant: third, reference angle: $\frac{\pi}{3}$