If an angle in standard position has its terminal side passing through the point $(-3,4)$, what is the measure of the angle in degrees?

• A. $228°$
• B. $246°$
• C. $123°$
• D. $114°$

Answer: (C)

Verified step by step guidance

1. Identify the coordinates of the point through which the terminal side of the angle passes. Here, the point is \((-3, 4)\).
2. Calculate the reference angle \(\theta_r\) by finding the angle the terminal side makes with the x-axis using the tangent function: \(\tan \theta_r = \left|\frac{y}{x}\right| = \left|\frac{4}{-3}\right| = \frac{4}{3}\).
3. Find the reference angle \(\theta_r\) by taking the arctangent (inverse tangent) of \(\frac{4}{3}\): \(\theta_r = \arctan\left(\frac{4}{3}\right)\).
4. Determine the quadrant in which the point \((-3, 4)\) lies. Since \(x\) is negative and \(y\) is positive, the point is in the second quadrant.
5. Calculate the actual angle \(\theta\) in standard position by subtracting the reference angle from \(180^\circ\) because angles in the second quadrant are \(180^\circ - \theta_r\): \(\theta = 180^\circ - \theta_r\).