Question

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

Explanation:

Step1: Recall Trigonometric Ratios

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

• 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.
• Tangent of \( x \) (\(\tan x\)) is defined as the ratio of the length of the opposite side to the length of the adjacent side.

Step2: Identify Sides Relative to \( x \)

• Opposite side to \( x \): \( c \) (does not form angle \( x \))
• Adjacent side to \( x \): \( a \) (forms angle \( x \) with the hypotenuse)
• Hypotenuse: \( b \) (longest side, opposite the right angle)

Step3: Apply Ratios

• \(\sin x = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{c}{b}\)
• \(\cos x = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{a}{b}\)
• \(\tan x = \frac{\text{opposite}}{\text{adjacent}} = \frac{c}{a}\)

Answer:

$\sin x = \frac{c}{b}$, $\tan x = \frac{c}{a}$, $\cos x = \frac{a}{b}$