Question

a right triangle has side lengths d, e, and f as shown below. use these lengths to find \\(\cos x\\), \\(\sin x\\), and \\(\tan x\\).

Explanation:

Step1: Recall Trigonometric Ratios

In a right triangle, for an acute angle \( x \):

• Cosine of \( x \) (\(\cos x\)) is defined as the ratio of the adjacent side to the hypotenuse.
• Sine of \( x \) (\(\sin x\)) is defined as the ratio of the opposite side to the hypotenuse.
• Tangent of \( x \) (\(\tan x\)) is defined as the ratio of the opposite side to the adjacent side.

Step2: Identify Sides Relative to \( x \)

• Adjacent side to \( x \): The side next to \( x \) (not the hypotenuse), which is \( d \).
• Opposite side to \( x \): The side opposite \( x \), which is \( e \).
• Hypotenuse: The longest side (opposite the right angle), which is \( f \).

Step3: Apply the Ratios

• For \(\cos x\):

\(\cos x = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{d}{f}\)

• For \(\sin x\):

\(\sin x = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{e}{f}\)

• For \(\tan x\):

\(\tan x = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{e}{d}\)

Answer:

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