Math / Geometry

Questionb.

Studdy Solution

STEP 1

1. The triangle is a right triangle.
2. One angle is $47^\circ$ .
3. The side opposite the $47^\circ$ angle is $12$ .
4. We need to find the lengths of the other two sides, $x$ and $y$ .

STEP 2

1. Identify the trigonometric functions that relate the given angle and side to the unknown sides.
2. Use the sine, cosine, or tangent functions to find the lengths of the unknown sides.
3. Solve the equations to find the values of $x$ and $y$ .

STEP 3

Identify the trigonometric functions that relate the given angle to the unknown sides. In a right triangle, we can use the sine, cosine, and tangent functions. The side opposite the $47^\circ$ angle is $12$ , and we need to find $x$ (adjacent side) and $y$ (hypotenuse).

STEP 4

Use the sine function to find the hypotenuse $y$ . The sine function is defined as:
$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$
For the $47^\circ$ angle:
$\sin(47^\circ) = \frac{12}{y}$

STEP 5

Solve for $y$ :
$y = \frac{12}{\sin(47^\circ)}$

STEP 6

Use the cosine function to find the adjacent side $x$ . The cosine function is defined as:
$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$
For the $47^\circ$ angle:
$\cos(47^\circ) = \frac{x}{y}$

STEP 7

Substitute the expression for $y$ from STEP_3 into the cosine equation:
$\cos(47^\circ) = \frac{x}{\frac{12}{\sin(47^\circ)}}$

STEP 8

Solve for $x$ :
$x = \cos(47^\circ) \cdot \frac{12}{\sin(47^\circ)}$

STEP 9

Calculate the values of $x$ and $y$ using a calculator:
$y = \frac{12}{\sin(47^\circ)} \approx 16.58$ $x = \cos(47^\circ) \cdot \frac{12}{\sin(47^\circ)} \approx 11.19$
The lengths of the sides are approximately:
$x \approx 11.19$ $y \approx 16.58$

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