Question

Find the value of x. Round to the nearest tenth.
Imagine a right triangle. The left slanted line is labelled 20, and the bottom line is labelled 9. Inside the triangle at the bottom left corner is "x degrees"

Answer 1

To find the value of \( x \) (the angle in degrees), we can use the sine function, cosine function, or tangent function. We'll use the sine function because we have the length of the opposite side (bottom line) and the hypotenuse (the slanted line).

Given:
- Hypotenuse = 20
- Adjacent side = 9
- Opposite side = \( a = \sqrt{20^2 - 9^2} \)

First, let's calculate the length of the opposite side using the Pythagorean theorem:

\[
a^2 + 9^2 = 20^2
\]

Where \( a \) is the length of the opposite side:

\[
a^2 + 81 = 400
\]
\[
a^2 = 400 - 81
\]
\[
a^2 = 319
\]
\[
a = \sqrt{319}
\]

Now we can calculate the angle \( x \) using the sine function:

\[
\sin(x) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\sqrt{319}}{20}
\]

To find \( x \):

\[
x = \arcsin\left(\frac{\sqrt{319}}{20}\right)
\]

Now we calculate \( \sqrt{319} \):

\[
\sqrt{319} \approx 17.8
\]

Now plug this value into the sine function:

\[
\sin(x) \approx \frac{17.8}{20} \approx 0.89
\]

Then we take the arcsine:

\[
x \approx \arcsin(0.89)
\]

Using a calculator to find \( \arcsin(0.89) \):

\[
x \approx 63.3^\circ
\]

Rounding to the nearest tenth, the value of \( x \) is:

\[
\boxed{63.3}
\]