If `costheta=(15)/(17)`, then find the value of `sintheta`.

Answer

To find the value of \(\sin \theta\) given that \(\cos \theta = \frac{15}{17}\), we can follow these steps:

### Step 1: Understand the relationship between sine and cosine
We know that in a right triangle, the cosine of an angle is defined as the ratio of the length of the adjacent side (base) to the length of the hypotenuse. Similarly, the sine of the angle is defined as the ratio of the length of the opposite side (perpendicular) to the hypotenuse.

### Step 2: Set up the triangle
Let’s denote the sides of the triangle:
- Let the base (adjacent side) be \(15x\).
- Let the hypotenuse be \(17x\).

### Step 3: Use the Pythagorean theorem
According to the Pythagorean theorem:
\[ \text{(Hypotenuse)}^2 = \text{(Base)}^2 + \text{(Perpendicular)}^2 \]
Substituting the known values:
\[ (17x)^2 = (15x)^2 + (AB)^2 \]
This simplifies to:
\[ 289x^2 = 225x^2 + (AB)^2 \]

### Step 4: Solve for the perpendicular side (AB)
Rearranging the equation gives:
\[ (AB)^2 = 289x^2 - 225x^2 \]
\[ (AB)^2 = 64x^2 \]
Taking the square root of both sides, we find:
\[ AB = 8x \]

### Step 5: Calculate \(\sin \theta\)
Now, we can find \(\sin \theta\):
\[ \sin \theta = \frac{\text{Perpendicular}}{\text{Hypotenuse}} = \frac{AB}{AC} = \frac{8x}{17x} \]
The \(x\) cancels out:
\[ \sin \theta = \frac{8}{17} \]

### Final Answer
Thus, the value of \(\sin \theta\) is \(\frac{8}{17}\).

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