Which equation can be used to determine the reference angle for an angle $\theta$ in standard position?

• A. $\alpha =\theta +90$
• B. $\alpha =\left|\theta \right|$
• C. $\alpha =|\theta -180|$
• D. $\alpha =360-\theta$

Answer: (C)

Verified step by step guidance

1. Understand that the reference angle \( \alpha \) is the acute angle formed between the terminal side of the given angle \( \theta \) in standard position and the x-axis.
2. Identify the quadrant in which the angle \( \theta \) lies, because the formula for the reference angle depends on the quadrant.
3. Recall the general formulas for reference angles based on the quadrant: for angles in the second quadrant, \( \alpha = |180^\circ - \theta| \); in the third quadrant, \( \alpha = |\theta - 180^\circ| \); and in the fourth quadrant, \( \alpha = |360^\circ - \theta| \).
4. Note that the absolute value ensures the reference angle is always positive and acute (less than 90 degrees).
5. From the given options, recognize that the correct formula for the reference angle \( \alpha \) when \( \theta \) is in the third quadrant is \( \alpha = |\theta - 180^\circ| \), which matches the correct answer.