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})? sin θ cos θ tan θ cot θ

Explanation:

Step1: Recall the unit - circle definitions

On the unit circle \(x = \cos\theta\) and \(y=\sin\theta\). For the point \((x_1,y_1)\) with central angle \(\theta\), \(x_1=\cos\theta\) and \(y_1 = \sin\theta\). For the point \((x_2,y_2)\) with central angle \(\pi-\theta\), we know that \(x_2=-\cos\theta\) and \(y_2=\sin\theta\) (using the identities \(\cos(\pi - \alpha)=-\cos\alpha\) and \(\sin(\pi-\alpha)=\sin\alpha\)). The ratio \(\frac{y_2}{x_2}=\frac{\sin\theta}{-\cos\theta}\).

Step2: Recall the trigonometric identities

We know that \(\tan\theta=\frac{\sin\theta}{\cos\theta}\). But if we consider the general form of trigonometric ratios, for a right - triangle (in the unit - circle context, we can think of the coordinates as the sides of a right - triangle with hypotenuse \(r = 1\)).
Another way: By the definition of the tangent function \(y = r\sin\theta\) and \(x = r\cos\theta\), \(\tan\theta=\frac{y}{x}\) (when \(x
eq0\)).

Answer:

\(\tan\theta\)