Math / Trigonometry

Question4) What is $\sin (x)$ ?

Studdy Solution

STEP 1

1. We are given a right triangle with an angle $x$ .
2. The hypotenuse of the triangle is 15.
3. The side adjacent to angle $x$ is 8.
4. We need to find $\sin(x)$ .

STEP 2

1. Recall the definition of sine in a right triangle.
2. Use the Pythagorean theorem to find the length of the opposite side.
3. Calculate $\sin(x)$ using the definition.

STEP 3

Recall that in a right triangle, the sine of an angle is defined as the ratio of the length of the opposite side to the hypotenuse.
$\sin(x) = \frac{\text{opposite}}{\text{hypotenuse}}$

STEP 4

Use the Pythagorean theorem to find the length of the opposite side. The Pythagorean theorem states:
$\text{hypotenuse}^2 = \text{adjacent}^2 + \text{opposite}^2$
Substitute the known values:
$15^2 = 8^2 + \text{opposite}^2$

STEP 5

Calculate the squares:
$225 = 64 + \text{opposite}^2$

STEP 6

Solve for the opposite side:
$\text{opposite}^2 = 225 - 64$ $\text{opposite}^2 = 161$ $\text{opposite} = \sqrt{161}$

STEP 7

Calculate $\sin(x)$ using the definition:
$\sin(x) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\sqrt{161}}{15}$
The value of $\sin(x)$ is:
$\boxed{\frac{\sqrt{161}}{15}}$

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