Question

the coordinate plane below shows a unit circle with central angle θ. which of these is equivalent to the ratio (\frac{y_{2}}{1})? (sinθ) ( - sinθ) (cosθ) ( - cosθ)

Explanation:

Step1: Recall the unit - circle definitions

On the unit circle \(x = \cos\theta\) and \(y=\sin\theta\). For a point \((x_1,y_1)\) on the unit circle with central angle \(\theta\), \(x_1=\cos\theta\) and \(y_1 = \sin\theta\). For a 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 \(A=\pi\) and \(B = \theta\). So \(\cos(\pi+\theta)=\cos\pi\cos\theta-\sin\pi\sin\theta=(- 1)\cos\theta-0\times\sin\theta=-\cos\theta\) and \(\sin(\pi+\theta)=\sin\pi\cos\theta+\cos\pi\sin\theta=0\times\cos\theta+(-1)\sin\theta=-\sin\theta\). So \(x_2=\cos(\pi + \theta)=-\cos\theta\) and \(y_2=\sin(\pi+\theta)=-\sin\theta\).

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

Since the radius of the unit circle \(r = 1\), and \(y_2\) is the \(y\) - coordinate of the point \((x_2,y_2)\). We know that \(y_2=\sin(\pi+\theta)\). Using the identity \(\sin(A + B)\) (where \(A=\pi\) and \(B=\theta\)) or the unit - circle symmetry. Another way: from the unit - circle, for a point \((x,y)\) on the unit circle \(y=\sin\alpha\) (where \(\alpha\) is the central angle). Here \(\alpha=\pi+\theta\), and \(\sin(\pi+\theta)=-\sin\theta\).

Answer:

\(-\sin\theta\)