Question

the coordinate plane below shows a unit circle with central angle θ. which of these is equivalent to the ratio ? sin θ - sin θ cos θ - cos θ

Explanation:

Step1: Recall the unit - circle definitions

On the unit circle \(x = \cos\alpha\) and \(y=\sin\alpha\), where \(\alpha\) is the central angle. For the point \((x_1,y_1)\) with central angle \(\theta\), we have \(x_1=\cos\theta\) and \(y_1 = \sin\theta\). For the point \((x_2,y_2)\) with central angle \(\pi-\theta\), we use the trigonometric identities \(\cos(A - B)=\cos A\cos B+\sin A\sin B\) and \(\sin(A - B)=\sin A\cos B-\cos A\sin B\). Here, \(\cos(\pi-\theta)=\cos\pi\cos\theta+\sin\pi\sin\theta=-\cos\theta\) and \(\sin(\pi - \theta)=\sin\pi\cos\theta-\cos\pi\sin\theta=\sin\theta\).

Step2: Analyze the ratio \(\frac{y_2}{1}\)

Since the radius of the unit - circle \(r = 1\), and for the point \((x_2,y_2)\) with central angle \(\pi-\theta\), \(y_2=\sin(\pi-\theta)\). By the co - function identity \(\sin(\pi-\theta)=\sin\theta\).

Answer:

\(\sin\theta\)