Question

#3 draw the angle and then find the measure of its reference angle.
a) 280°
b) -135°
c) 101°

Explanation:

Step1: Determine the quadrant of \(280^{\circ}\)

Since \(270^{\circ}<280^{\circ}<360^{\circ}\), \(280^{\circ}\) is in the fourth quadrant.

Step2: Calculate the reference angle for \(280^{\circ}\)

The formula for the reference angle \(\theta_{r}\) of an angle \(\theta\) in the fourth quadrant is \(\theta_{r} = 360^{\circ}-\theta\). So, \(\theta_{r}=360^{\circ} - 280^{\circ}=80^{\circ}\)

Step3: Deal with \(- 135^{\circ}\)

Add \(360^{\circ}\) to \(-135^{\circ}\) to get a co - terminal angle. \(-135^{\circ}+360^{\circ}=225^{\circ}\). Since \(180^{\circ}<225^{\circ}<270^{\circ}\), \(225^{\circ}\) (and thus \(-135^{\circ}\)) is in the third quadrant.

Step4: Calculate the reference angle for \(-135^{\circ}\)

The formula for the reference angle \(\theta_{r}\) of an angle \(\theta\) in the third quadrant is \(\theta_{r}=\theta - 180^{\circ}\). So, \(\theta_{r}=225^{\circ}-180^{\circ} = 45^{\circ}\)

Step5: Determine the quadrant of \(101^{\circ}\)

Since \(90^{\circ}<101^{\circ}<180^{\circ}\), \(101^{\circ}\) is in the second quadrant.

Step6: Calculate the reference angle for \(101^{\circ}\)

The formula for the reference angle \(\theta_{r}\) of an angle \(\theta\) in the second quadrant is \(\theta_{r}=180^{\circ}-\theta\). So, \(\theta_{r}=180^{\circ}-101^{\circ}=79^{\circ}\)

Answer:

a) The reference angle of \(280^{\circ}\) is \(80^{\circ}\)
b) The reference angle of \(-135^{\circ}\) is \(45^{\circ}\)
c) The reference angle of \(101^{\circ}\) is \(79^{\circ}\)