Question

i. for each angle θ below, do the following: a.) draw angle with a colored marker and make sure to include direction in your drawing. b.) if angle is more than ±360°, then indicate that in drawing with “swirlies”. 1.) θ = 310° 2.) θ=-125° 3.) θ = 510° 4.) θ=-700° 5.) θ=-1350° 6.) θ = \frac{5\pi}{4} 7.) θ=-\frac{\pi}{2} 8.) θ = \frac{25\pi}{6} 9.) θ=-\frac{10\pi}{3} 10.) θ = 7\pi
ii. complete the chart below about converting angle measures. show work on line! degree measure → radian measure radian measure → degree measure 11.) θ = 155°→ __________ 14.) θ = \frac{7\pi}{6}→\frac{7\pi}{6}\cdot\frac{180}{\pi} 12.) θ=-330°→ ________ 15.) θ = \frac{\pi}{3}→ ________ 13.) θ = 720°→ ________ 16.) θ=-\frac{26\pi}{15}→ __________
iii. find a positive and a negative coterminal angle for each given angle. show work! 17.) 114°→ positive coterminal angle = 474° negative coterminal angle = __________ 18.) -\frac{4\pi}{9}→ positive coterminal angle = ________ negative coterminal angle = __________
iv. determine the measure of each angle. keep units consistent. must show work! 19.) 20.) 21.) 22.)
v. determine the reference angle for each given angle θ. show your work! 23.) θ = 323°→ reference angle: __________ 24.) θ = 242°→ reference angle: ________ 25.) θ = 127°→ reference angle: ________ 26.) θ=-135°→ reference angle: ________ 27.) θ = 744°→ reference angle: ________ 28.) θ=-566°→ reference angle: __________

Explanation:

Step1: Recall degree - radian conversion formula

The formula to convert degrees to radians is $\theta_{rad}=\theta_{deg}\times\frac{\pi}{180}$, and to convert radians to degrees is $\theta_{deg}=\theta_{rad}\times\frac{180}{\pi}$.

Step2: Solve 11.

For $\theta = 155^{\circ}$, $\theta_{rad}=155\times\frac{\pi}{180}=\frac{31\pi}{36}$.

Step3: Solve 12.

For $\theta=- 330^{\circ}$, $\theta_{rad}=-330\times\frac{\pi}{180}=-\frac{11\pi}{6}$.

Step4: Solve 13.

For $\theta = 720^{\circ}$, $\theta_{rad}=720\times\frac{\pi}{180}=4\pi$.

Step5: Solve 14.

For $\theta=\frac{7\pi}{6}$, $\theta_{deg}=\frac{7\pi}{6}\times\frac{180}{\pi}=210^{\circ}$.

Step6: Solve 15.

For $\theta=\frac{\pi}{3}$, $\theta_{deg}=\frac{\pi}{3}\times\frac{180}{\pi}=60^{\circ}$.

Step7: Solve 16.

For $\theta =-\frac{26\pi}{15}$, $\theta_{deg}=-\frac{26\pi}{15}\times\frac{180}{\pi}=-312^{\circ}$.

Step8: Recall coterminal - angle formula

Coterminal angles of $\theta$ are given by $\theta\pm360^{\circ}n$ (in degrees) or $\theta\pm2\pi n$ (in radians), where $n$ is an integer.

Step9: Solve 17.

For $\theta = 114^{\circ}$, a positive coterminal angle: $\theta_{+}=114 + 360=474^{\circ}$, a negative coterminal angle: $\theta_{-}=114-360=-246^{\circ}$.

Step10: Solve 18.

For $\theta=-\frac{4\pi}{9}$, a positive coterminal angle: $\theta_{+}=-\frac{4\pi}{9}+2\pi=\frac{- 4\pi + 18\pi}{9}=\frac{14\pi}{9}$, a negative coterminal angle: $\theta_{-}=-\frac{4\pi}{9}-2\pi=\frac{-4\pi - 18\pi}{9}=-\frac{22\pi}{9}$.

Step11: Recall reference - angle rules

For an angle $\theta$ in standard position:

• If $\theta$ is in the first quadrant, the reference angle $\theta_{r}=\theta$.
• If $\theta$ is in the second quadrant, $\theta_{r}=180^{\circ}-\theta$ (in degrees) or $\pi-\theta$ (in radians).
• If $\theta$ is in the third quadrant, $\theta_{r}=\theta - 180^{\circ}$ (in degrees) or $\theta-\pi$ (in radians).
• If $\theta$ is in the fourth quadrant, $\theta_{r}=360^{\circ}-\theta$ (in degrees) or $2\pi-\theta$ (in radians).

Step12: Solve 23.

For $\theta = 323^{\circ}$ (fourth - quadrant), $\theta_{r}=360 - 323=37^{\circ}$.

Step13: Solve 24.

For $\theta = 242^{\circ}$ (third - quadrant), $\theta_{r}=242 - 180 = 62^{\circ}$.

Step14: Solve 25.

For $\theta = 127^{\circ}$ (second - quadrant), $\theta_{r}=180 - 127=53^{\circ}$.

Step15: Solve 26.

For $\theta=-135^{\circ}$, add $360^{\circ}$ to get it in the range $[0^{\circ},360^{\circ}]$, $\theta=-135 + 360 = 225^{\circ}$ (third - quadrant), $\theta_{r}=225 - 180=45^{\circ}$.

Step16: Solve 27.

For $\theta = 744^{\circ}$, subtract $2\times360^{\circ}$ to get $\theta = 744-720 = 24^{\circ}$ (first - quadrant), $\theta_{r}=24^{\circ}$.

Step17: Solve 28.

For $\theta=-566^{\circ}$, add $2\times360^{\circ}=720^{\circ}$, $\theta=-566 + 720 = 154^{\circ}$ (second - quadrant), $\theta_{r}=180 - 154 = 26^{\circ}$.

Answer:

1. $\frac{31\pi}{36}$
2. $-\frac{11\pi}{6}$
3. $4\pi$
4. $210^{\circ}$
5. $60^{\circ}$
6. $-312^{\circ}$
7. Positive: $474^{\circ}$, Negative: $-246^{\circ}$
8. Positive: $\frac{14\pi}{9}$, Negative: $-\frac{22\pi}{9}$
9. $37^{\circ}$
10. $62^{\circ}$
11. $53^{\circ}$
12. $45^{\circ}$
13. $24^{\circ}$
14. $26^{\circ}$