Question

find an angle in each quadrant with a common reference angle with 337°, from 0°≤θ<360°
answer
attempt 1 out of 2
quadrant i: °
quadrant ii: °
quadrant iii: °
quadrant iv: °

Explanation:

Step1: Calculate the reference angle of \(337^{\circ}\)

The reference angle formula for an angle \(\theta\) in the fourth - quadrant (\(270^{\circ}<\theta < 360^{\circ}\)) is \(\text{Reference angle}=360^{\circ}-\theta\).
For \(\theta = 337^{\circ}\), the reference angle \(\alpha=360^{\circ}-337^{\circ}=23^{\circ}\)

Step2: Find the angle in Quadrant I

In Quadrant I, the angle is equal to the reference angle. So, \(\theta_1 = 23^{\circ}\)

Step3: Find the angle in Quadrant II

In Quadrant II, the angle formula is \(\theta=180^{\circ}-\alpha\). Substituting \(\alpha = 23^{\circ}\), we get \(\theta_2=180^{\circ}-23^{\circ}=157^{\circ}\)

Step4: Find the angle in Quadrant III

In Quadrant III, the angle formula is \(\theta = 180^{\circ}+\alpha\). Substituting \(\alpha=23^{\circ}\), we get \(\theta_3=180^{\circ}+23^{\circ}=203^{\circ}\)

Step5: Find the angle in Quadrant IV

We already know that for \(\theta = 337^{\circ}\) (which is in Quadrant IV), we used the reference - angle formula \(\alpha=360^{\circ}-\theta\). But if we start from the reference angle \(\alpha = 23^{\circ}\), the angle in Quadrant IV is \(\theta_4=360^{\circ}-23^{\circ}=337^{\circ}\)

Answer:

Quadrant I: \(23^{\circ}\)
Quadrant II: \(157^{\circ}\)
Quadrant III: \(203^{\circ}\)
Quadrant IV: \(337^{\circ}\)