If `P(-3, 4)` is a point on the terminal side of angle `theta`, what is the value of `costheta`?

Options

A. `-(3)/(4)`
B. `-(3)/(5)`
C. `(3)/(4)`
D. `(4)/(5)`

Answer

To find the value of \( \cos \theta \) for the point \( P(-3, 4) \) on the terminal side of angle \( \theta \), we can follow these steps:

### Step 1: Identify the coordinates
The coordinates of the point are \( P(-3, 4) \). Here, the x-coordinate (base) is \( -3 \) and the y-coordinate (perpendicular) is \( 4 \).

### Step 2: Calculate the hypotenuse
To find the hypotenuse \( H \), we use the Pythagorean theorem:
\[ H^2 = x^2 + y^2 \]
Substituting the values:
\[ H^2 = (-3)^2 + (4)^2 \]
Calculating the squares:
\[ H^2 = 9 + 16 = 25 \]
Taking the square root to find \( H \):
\[ H = \sqrt{25} = 5 \]

### Step 3: Calculate \( \cos \theta \)
The cosine of the angle \( \theta \) is given by the ratio of the adjacent side (base) to the hypotenuse:
\[ \cos \theta = \frac{\text{base}}{\text{hypotenuse}} = \frac{-3}{5} \]

### Final Answer
Thus, the value of \( \cos \theta \) is:
\[ \cos \theta = -\frac{3}{5} \]

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