If $7\cot P=24$, then find $\sin P$.

The correct option is $\frac{7}{25}$.

To find the value of $\sin P$ from the given equation, let us break down the solution step-by-step.

Step 1: Simplify the given equation to find $\cot P$.
We are given:

$7\cot P=24$

Dividing both sides by 7, we get:

$\cot P=\frac{24}{7}$

Step 2: Understand the trigonometric ratio.
In a right-angled triangle, the cotangent ratio ($\cot P$) is defined as the ratio of the length of the side adjacent to angle P to the side opposite to angle P:

$\cot P=\frac{\text{Adjacent Side}}{\text{Opposite Side}}=\frac{24}{7}$

Therefore, we can assume the lengths of the sides as:
Adjacent side = $24$
Opposite side = $7$

Step 3: Calculate the hypotenuse using the Pythagorean theorem.
According to the Pythagorean theorem:

${\text{Hypotenuse}}^2={\text{Opposite}}^2+{\text{Adjacent}}^2$

Substituting the side values:

${\text{Hypotenuse}}^2=7^2+{24}^2$

${\text{Hypotenuse}}^2=49+576$

${\text{Hypotenuse}}^2=625$

Taking the square root on both sides:

$\text{Hypotenuse}=\sqrt{625}=25$

Step 4: Find the value of $\sin P$.
The sine ratio ($\sin P$) is defined as the ratio of the opposite side to the hypotenuse:

$\sin P=\frac{\text{Opposite Side}}{\text{Hypotenuse}}$

Substituting the values we found:

$\sin P=\frac{7}{25}$