Question

look at the figure:

if \\(\tan x^\circ = \frac{3}{y}\\) and \\(\cos x^\circ = \frac{y}{z}\\), what is the value of \\(\sin x^\circ\\)? (1 point)

\\(\bigcirc\\) \\(\sin x^\circ = \frac{3}{z}\\)

\\(\bigcirc\\) \\(\sin x^\circ = 3y\\)

\\(\bigcirc\\) \\(\sin x^\circ = \frac{z}{3}\\)

\\(\bigcirc\\) \\(\sin x^\circ = 3z\\)

Explanation:

Identify trigonometric definitions

We use Right Triangle Trigonometry to relate the ratios.
Let the side opposite to \(x^\circ\) be \(a\), the side adjacent to \(x^\circ\) be \(b\), and the hypotenuse be \(c\).
The trigonometric ratios are defined as:

$$ \tan x^\circ = \frac{\text{opposite}}{\text{adjacent}} = \frac{a}{b} $$

$$ \cos x^\circ = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{b}{c} $$

$$ \sin x^\circ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{c} $$

Map given values to triangle sides

We are given:

$$ \tan x^\circ = \frac{3}{y} $$

Comparing this to \(\frac{\text{opposite}}{\text{adjacent}}\), we can set:

• Opposite side \(a = 3\)
• Adjacent side \(b = y\)

We are also given:

$$ \cos x^\circ = \frac{y}{z} $$

Comparing this to \(\frac{\text{adjacent}}{\text{hypotenuse}}\) where \(b = y\), we find:

• Hypotenuse \(c = z\)

Calculate the sine ratio

Using the Sine Ratio definition:

$$ \sin x^\circ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{c} $$

Substituting the values \(a = 3\) and \(c = z\):

$$ \sin x^\circ = \frac{3}{z} $$

Answer:

• (A) \(\sin x^\circ = \frac{3}{z}\) (Correct answer)
• (B) \(\sin x^\circ = 3y\)
• (C) \(\sin x^\circ = \frac{z}{3}\)
• (D) \(\sin x^\circ = 3z\)