If $\theta$ is in standard position and its terminal side passes through the point $(2,3)$, what is the degree measure of $\theta$ rounded to the nearest whole number?

• A. $36$
• B. $73$
• C. $56$
• D. $65$

Answer: (C)

Verified step by step guidance

1. Identify the angle \( \theta \) in standard position whose terminal side passes through the point \( (2, 3) \). This means the point lies on the terminal side of the angle measured from the positive x-axis.
2. Calculate the reference angle by finding the arctangent of the ratio of the y-coordinate to the x-coordinate of the point. Use the formula: \[ \theta = \tan^{-1} \left( \frac{y}{x} \right) = \tan^{-1} \left( \frac{3}{2} \right) \]
3. Evaluate the arctangent expression to find the angle in degrees. This will give the measure of the angle \( \theta \) in the first quadrant since both x and y are positive.
4. Round the resulting angle to the nearest whole number to get the degree measure of \( \theta \).
5. Verify that the angle is in the correct quadrant based on the signs of x and y. Since both are positive, the angle is indeed in the first quadrant, so no further adjustment is needed.