Question

enter the coordinates of the point on the unit circle at the given angle. 270° (?, )

Explanation:

Step1: Recall the unit - circle coordinate formula

For a point \((x,y)\) on the unit circle at an angle \(\theta\), \(x = \cos\theta\) and \(y=\sin\theta\).

Step2: Calculate \(x\) - coordinate

Given \(\theta = 270^{\circ}\), \(x=\cos(270^{\circ})\). Since \(\cos(270^{\circ})=\cos(360^{\circ}- 90^{\circ})=\cos(90^{\circ}) = 0\) (using the cosine subtraction formula \(\cos(A - B)=\cos A\cos B+\sin A\sin B\) with \(A = 360^{\circ}\), \(B=90^{\circ}\), \(\cos(360^{\circ}) = 1\), \(\sin(360^{\circ})=0\), \(\cos(90^{\circ}) = 0\), \(\sin(90^{\circ})=1\)).

Step3: Calculate \(y\) - coordinate

\(y = \sin(270^{\circ})\). Since \(\sin(270^{\circ})=\sin(360^{\circ}-90^{\circ})=\sin(360^{\circ})\cos(90^{\circ})-\cos(360^{\circ})\sin(90^{\circ})\) (using the sine subtraction formula \(\sin(A - B)=\sin A\cos B-\cos A\sin B\) with \(A = 360^{\circ}\), \(B = 90^{\circ}\)), and \(\sin(360^{\circ}) = 0\), \(\cos(360^{\circ})=1\), \(\cos(90^{\circ}) = 0\), \(\sin(90^{\circ})=1\), we get \(y=- 1\).

Answer:

\((0,-1)\)