Question
Name the reference angle for \( 315^{\circ} \). \( \hat{\theta}=\square \)
Ask by Ramirez Schultz. in the United States
Jan 20,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The reference angle for \( 315^{\circ} \) is \( 45^{\circ} \).
Solution
The reference angle for \( 315^{\circ} \) is calculated by finding the acute angle that \( 315^{\circ} \) makes with the nearest part of the x-axis.
Since \( 315^{\circ} \) is in the **fourth quadrant**, the reference angle \( \hat{\theta} \) is:
\[
\hat{\theta} = 360^{\circ} - 315^{\circ} = 45^{\circ}
\]
**Answer:**
\[
\hat{\theta} = 45^{\circ}
\]
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Simplify this solution Mind Expander
The reference angle for \( 315^{\circ} \) can be found by subtracting it from \( 360^{\circ} \). So, \( 360^{\circ} - 315^{\circ} = 45^{\circ} \). Therefore, the reference angle \( \hat{\theta} = 45^{\circ} \). Additionally, \( 315^{\circ} \) lies in the fourth quadrant where angles are positive but their corresponding reference angles are measured as positive acute angles. This means that while the angle itself is greater than \( 270^{\circ} \), the reference angle helps simplify calculations in trigonometry, such as finding sine, cosine, and tangent values.
