Question

2.
there is a right triangle with a right angle, one angle is 22°, the opposite side of 22° is 6, the hypotenuse is x, and the adjacent side of 22° is y.

Explanation:

Step1: Identify trigonometric ratios

We have a right - triangle with an angle of \(22^{\circ}\), the opposite side to the angle is \(6\), the hypotenuse is \(x\), and the adjacent side is \(y\). We can use the sine function to find \(x\) and the tangent function to find \(y\) (or we can also use cosine and Pythagorean theorem, but sine and tangent are more direct here).

The sine of an angle \(\theta\) in a right - triangle is given by \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\) and the tangent of an angle \(\theta\) is given by \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\)

Step2: Find the value of \(x\)

Given \(\theta = 22^{\circ}\), opposite side \(= 6\), hypotenuse \(=x\)

Using the sine formula \(\sin(22^{\circ})=\frac{6}{x}\)

We can solve for \(x\) by cross - multiplying: \(x=\frac{6}{\sin(22^{\circ})}\)

We know that \(\sin(22^{\circ})\approx0.3746\)

So \(x=\frac{6}{0.3746}\approx16.02\)

Step3: Find the value of \(y\)

Using the tangent formula \(\tan(22^{\circ})=\frac{6}{y}\)

We can solve for \(y\) by cross - multiplying: \(y = \frac{6}{\tan(22^{\circ})}\)

We know that \(\tan(22^{\circ})\approx0.4040\)

So \(y=\frac{6}{0.4040}\approx14.85\)

Answer:

If we want to find \(x\): \(x\approx\boldsymbol{16.02}\) (if we want to find \(y\): \(y\approx\boldsymbol{14.85}\)) (depending on which variable we are asked to find, but since the problem is about the right - triangle with angle \(22^{\circ}\), opposite side \(6\), we can calculate both. If we assume we need to find \(x\) first, the answer for \(x\) is approximately \(16.02\) and for \(y\) is approximately \(14.85\))