Question

find \\(\tan\theta\\), where \\(\theta\\) is the angle shown. give an exact value, not a decimal approximation. the right triangle has hypotenuse 25, one leg 7, and angle \\(\theta\\) at the vertex with the leg of length 7 and the hypotenuse. \\(\tan\theta = \square\\)

Explanation:

Step1: Recall tangent definition

In a right triangle, $\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$. First, find the opposite side using Pythagoras. Let opposite side be $y$, hypotenuse $25$, adjacent $7$. So $y = \sqrt{25^2 - 7^2} = \sqrt{625 - 49} = \sqrt{576} = 24$.

Step2: Calculate $\tan\theta$

Now, $\tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{24}{7}$.

Answer:

$\frac{24}{7}$