If an angle $\angle ABD$ is in standard position and its terminal side passes through the point $(3,4)$, what is the measure of $\angle ABD$ to the nearest degree?

• A. $114°$
• B. $124°$
• C. $27°$
• D. $53°$

Answer: (D)

Verified step by step guidance

1. Identify that the angle \( \angle ABD \) is in standard position, meaning its vertex is at the origin and its initial side lies along the positive x-axis.
2. Recognize that the terminal side of the angle passes through the point \( (3, 4) \). This point gives the coordinates \( x = 3 \) and \( y = 4 \).
3. Calculate the reference angle \( \theta \) using the tangent function, since \( \tan(\theta) = \frac{y}{x} \). So, \( \theta = \arctan\left(\frac{4}{3}\right) \).
4. Use the arctangent value to find the angle in degrees. Remember to convert from radians to degrees if necessary by multiplying by \( \frac{180}{\pi} \).
5. Since the point \( (3, 4) \) lies in the first quadrant (both x and y positive), the angle \( \angle ABD \) is simply the reference angle \( \theta \) found above.