Question

the coordinate plane below shows a unit circle with central angle θ.which of these is equivalent to the ratio \\( \frac{y_2}{x_2} \\)?sin θcos θtan θcot θ

Explanation:

Step1: Recall the unit - circle definitions

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

Step2: Calculate the ratio \(\frac{y_2}{x_2}\)

Substitute \(x_2 =-\cos\theta\) and \(y_2=-\sin\theta\) into the ratio \(\frac{y_2}{x_2}\). Then \(\frac{y_2}{x_2}=\frac{-\sin\theta}{-\cos\theta}\)

Step3: Simplify the ratio

Cancel out the negative signs. \(\frac{-\sin\theta}{-\cos\theta}=\frac{\sin\theta}{\cos\theta}\)

Step4: Use the tangent identity

Recall the identity \(\tan\alpha=\frac{\sin\alpha}{\cos\alpha}\). When \(\alpha = \theta\), \(\frac{\sin\theta}{\cos\theta}=\tan\theta\)

Answer:

\(\tan\theta\) (the third option)