Math / Trigonometry

QuestionFind the value of $\cos C$ rounded to the nearest hundredth, if necessary.

Studdy Solution

STEP 1

1. The triangle $\triangle CDE$ is a right triangle with a right angle at $D$ .
2. The side $CD$ is the adjacent side to angle $C$ .
3. The side $DE$ is the opposite side to angle $C$ .
4. We need to find $\cos C$ , which is the ratio of the adjacent side to the hypotenuse.

STEP 2

1. Identify the sides of the triangle.
2. Use the Pythagorean theorem to find the hypotenuse.
3. Calculate $\cos C$ .

STEP 3

Identify the sides of the triangle: - $CD = 4$ units (adjacent to angle $C$ ) - $DE = 8$ units (opposite to angle $C$ )

STEP 4

Use the Pythagorean theorem to find the hypotenuse $CE$ . The Pythagorean theorem states:
$CD^2 + DE^2 = CE^2$
Substitute the known values:
$4^2 + 8^2 = CE^2$

STEP 5

Calculate the squares:
$16 + 64 = CE^2$ $80 = CE^2$

STEP 6

Find $CE$ by taking the square root:
$CE = \sqrt{80}$ $CE = \sqrt{16 \times 5}$ $CE = 4\sqrt{5}$

STEP 7

Calculate $\cos C$ using the formula $\cos C = \frac{\text{adjacent}}{\text{hypotenuse}}$ :
$\cos C = \frac{CD}{CE}$ $\cos C = \frac{4}{4\sqrt{5}}$

STEP 8

Simplify the expression:
$\cos C = \frac{1}{\sqrt{5}}$
To rationalize the denominator:
$\cos C = \frac{1}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}}$ $\cos C = \frac{\sqrt{5}}{5}$

STEP 9

Calculate the decimal value and round to the nearest hundredth:
$\cos C \approx \frac{\sqrt{5}}{5} \approx 0.447$
Rounded to the nearest hundredth:
$\cos C \approx 0.45$
The value of $\cos C$ rounded to the nearest hundredth is:
$\boxed{0.45}$

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