A right triangle has side lengths d, e, and f as shown below.Use these lengths to find tanx, cosx, and sinx.xfed

1. Break Down the Problem

To find the trigonometric ratios $\tan x$, $\cos x$, and $\sin x$ for the right triangle with side lengths $d$, $e$, and $f$, we first need to identify the angles based on the sides provided:

• $d$ is the length of the side opposite angle $x$.
• $e$ is the length of the side adjacent to angle $x$.
• $f$ is the length of the hypotenuse.

2. Relevant Concepts

The definitions of the trigonometric ratios based on the sides of a right triangle are as follows:

• Tangent: $\tan x = \frac{\text{opposite}}{\text{adjacent}} = \frac{d}{e}$
• Cosine: $\cos x = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{e}{f}$
• Sine: $\sin x = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{d}{f}$

3. Analysis and Detail

1. Calculate $\tan x$: $\tan x = \frac{d}{e}$

2. Calculate $\cos x$: $\cos x = \frac{e}{f}$

3. Calculate $\sin x$: $\sin x = \frac{d}{f}$

4. Verify and Summarize

Each of the trigonometric ratios are defined based on the relationships of the sides of a right triangle. Thus, the calculations for $\tan x$, $\cos x$, and $\sin x$ have been determined correctly based on the triangle's dimensions.

Final Answer

$\tan x = \frac{d}{e}, \quad \cos x = \frac{e}{f}, \quad \sin x = \frac{d}{f}$