Question
For the given angle \( \theta \), identify the reference angle, \( \varphi \) \( \theta=\frac{11 \pi}{6} \quad \varphi= \) ? \( \frac{\pi}{6} \) \( \frac{\pi}{3} \) \( 2 \pi \) DONE
Ask by May Nichols. in the United States
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The reference angle is \( \frac{\pi}{6} \).
Solution
To find the reference angle (ϕ) for a given angle θ, we use the fact that the reference angle is the acute angle that the terminal side of θ makes with the x-axis.
Given:
θ = 11π/6
Since 11π/6 is in the fourth quadrant, the reference angle is calculated as:
ϕ = 2π − θ
= 2π − 11π/6
= (12π/6 − 11π/6)
= π/6
Thus, the reference angle is π/6.
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Simplify this solution The Deep Dive
To find the reference angle \( \varphi \) for \( \theta = \frac{11\pi}{6} \), we must first determine the position of \( \theta \) on the unit circle. Since \( \frac{11\pi}{6} \) is in the fourth quadrant, we find the reference angle by subtracting it from \( 2\pi \): \[ \varphi = 2\pi - \frac{11\pi}{6} = \frac{12\pi}{6} - \frac{11\pi}{6} = \frac{\pi}{6}. \] Therefore, the reference angle \( \varphi \) is \( \frac{\pi}{6} \).
