In the context of angles in standard position, if point C is at $(0,0)$, point O is at $(1,0)$, and point A is at $(0,1)$, what is the measure of angle COA in degrees?

• A. $180°$
• B. $45°$
• C. $0°$
• D. $90°$

Answer: (D)

Verified step by step guidance

1. Identify the points given: C(0,0), O(1,0), and A(0,1). The angle COA is the angle formed at point O by the points C and A.
2. Find the vectors \( \overrightarrow{OC} \) and \( \overrightarrow{OA} \) originating from point O. Calculate these by subtracting the coordinates of O from C and A respectively: \( \overrightarrow{OC} = (0 - 1, 0 - 0) = (-1, 0) \) and \( \overrightarrow{OA} = (0 - 1, 1 - 0) = (-1, 1) \).
3. Use the dot product formula to find the cosine of the angle between the two vectors: \( \overrightarrow{OC} \cdot \overrightarrow{OA} = |\overrightarrow{OC}| \times |\overrightarrow{OA}| \times \cos(\theta) \).
4. Calculate the dot product \( \overrightarrow{OC} \cdot \overrightarrow{OA} = (-1)(-1) + (0)(1) = 1 \). Then find the magnitudes: \( |\overrightarrow{OC}| = \sqrt{(-1)^2 + 0^2} = 1 \) and \( |\overrightarrow{OA}| = \sqrt{(-1)^2 + 1^2} = \sqrt{2} \).
5. Solve for \( \cos(\theta) \) using \( \cos(\theta) = \frac{\overrightarrow{OC} \cdot \overrightarrow{OA}}{|\overrightarrow{OC}| \times |\overrightarrow{OA}|} = \frac{1}{1 \times \sqrt{2}} = \frac{1}{\sqrt{2}} \). Then find \( \theta \) by taking the inverse cosine (arccos) of \( \frac{1}{\sqrt{2}} \), which corresponds to the angle COA.