If point `P(-5,12)` lies on the terminal side of `angletheta` in standard position, `sintheta`=

Options

A. `-(12)/(13)`
B. `(-5)/(12)`
C. `(-5)/(13)`
D. `(12)/(13)`

Explanation

To find \(\sin \theta\) for the angle \(\theta\) whose terminal side passes through the point \(P(-5, 12)\), we can follow these steps:

### Step 1: Identify the coordinates
The point \(P\) has coordinates \((-5, 12)\). Here, the x-coordinate is \(-5\) and the y-coordinate is \(12\).

### Step 2: Determine the radius (hypotenuse)
Using the Pythagorean theorem, we can find the radius (hypotenuse) \(r\) of the right triangle formed by the point \(P\) and the origin \(O(0, 0)\). The formula is:
\[ r = \sqrt{x^2 + y^2} \]
Substituting the values:
\[ r = \sqrt{(-5)^2 + (12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]

### Step 3: Calculate \(\sin \theta\)
The sine of the angle \(\theta\) is defined as the ratio of the length of the opposite side (y-coordinate) to the length of the hypotenuse:
\[ \sin \theta = \frac{y}{r} \]
Substituting the values we found:
\[ \sin \theta = \frac{12}{13} \]

### Final Answer
Thus, the value of \(\sin \theta\) is:
\[ \sin \theta = \frac{12}{13} \]

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