Question

a right triangle has side lengths d, e, and f as shown below. use these lengths to find tanx, sinx, and cosx. tanx = sinx = cosx =

Explanation:

Step1: Recall Trigonometric Ratios

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

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

For angle \( x \) in the given right triangle:

• Opposite side to \( x \) is \( f \)
• Adjacent side to \( x \) is \( e \)
• Hypotenuse is \( d \)

Step2: Calculate \( \tan x \)

Using the definition of tangent:
\( \tan x=\frac{\text{opposite}}{\text{adjacent}}=\frac{f}{e} \)

Step3: Calculate \( \sin x \)

Using the definition of sine:
\( \sin x=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{f}{d} \)

Step4: Calculate \( \cos x \)

Using the definition of cosine:
\( \cos x=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{e}{d} \)

Answer:

\( \tan x = \boldsymbol{\frac{f}{e}} \)
\( \sin x = \boldsymbol{\frac{f}{d}} \)
\( \cos x = \boldsymbol{\frac{e}{d}} \)