Math / Geometry

Question5. 6. 7.

Studdy Solution

STEP 1

What is this asking? We need to find the length of the hypotenuse ( $x$ ) for three right-angled triangles, given one angle and one side length. Watch out! Make sure your calculator is in degree mode, not radians!
Also, remember SOH CAH TOA.

STEP 2

1. Solve Triangle 5
2. Solve Triangle 6
3. Solve Triangle 7

STEP 3

Alright, triangle number 5!
We've got an angle of 48 degrees, a side length of 27, and we're trying to find the hypotenuse, which is labeled $x$ .

STEP 4

Since we have the angle, the adjacent side, and we want the hypotenuse, we'll use cosine!
Remember, CAH: Cosine is Adjacent over Hypotenuse.

STEP 5

So, we can set up the equation: $\cos(48^\circ) = \frac{27}{x}$ .

STEP 6

To solve for $x$ , we can multiply both sides by $x$ and then divide both sides by $\cos(48^\circ)$ .
This gives us $x = \frac{27}{\cos(48^\circ)}$ .

STEP 7

Now, grab your calculator and make sure it's in degree mode!
Calculate the value: $x = \frac{27}{\cos(48^\circ)} \approx \frac{27}{\textbf{0.6691}} \approx \textbf{40.35}$ .
So, $x$ is approximately 40.35.

STEP 8

On to triangle 6!
This time we have an angle of 19 degrees, a side length of 6, and we're looking for the hypotenuse $x$ .

STEP 9

Again, we have the angle, the adjacent side, and we want the hypotenuse, so it's cosine time!
CAH!

STEP 10

Our equation is: $\cos(19^\circ) = \frac{6}{x}$ .

STEP 11

Solving for $x$ gives us $x = \frac{6}{\cos(19^\circ)}$ .

STEP 12

Calculator time! $x = \frac{6}{\cos(19^\circ)} \approx \frac{6}{\textbf{0.9455}} \approx \textbf{6.35}$ . $x$ is approximately 6.35.

STEP 13

Last one!
Triangle 7 has an angle of 52 degrees, a side of 45 m, and we're finding the hypotenuse $x$ .

STEP 14

Yep, you guessed it, cosine again!
We have the angle, the adjacent side, and we need the hypotenuse.
CAH is our friend!

STEP 15

The equation is $\cos(52^\circ) = \frac{45}{x}$ .

STEP 16

Solving for $x$ gives us $x = \frac{45}{\cos(52^\circ)}$ .

STEP 17

Time for the final calculation! $x = \frac{45}{\cos(52^\circ)} \approx \frac{45}{\textbf{0.6157}} \approx \textbf{73.08}$ . $x$ is approximately 73.08 m.

STEP 18

Triangle 5: $x \approx 40.35$ Triangle 6: $x \approx 6.35$ Triangle 7: $x \approx 73.08$ m

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