Question

as shown in the standard $(x,y)$ coordinate plane above, an angle with a vertex located at the origin and a measure of $\theta$ passing through $(-5,-12)$ and the negative $x$-axis. what is the value of the sine $\theta$? a $-\frac{12}{13}$ b $-\frac{5}{12}$ c $-\frac{5}{13}$ d $\frac{12}{13}$ e 13

Explanation:

Step1: Calculate the radius \( r \)

Use the formula \( r=\sqrt{x^{2}+y^{2}} \) for a point \((x,y)\). Here \(x = - 5\) and \(y=-12\).

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Step2: Use the definition of sine function

The definition of \(\sin\theta=\frac{y}{r}\). Substitute \(y=-12\) and \(r = 13\) into the formula.

$$ \sin\theta=\frac{-12}{13} $$

Answer:

A. \(-\frac{12}{13}\)