Math / Geometry

QuestionSolve for $x$ . Round to the nearest tenth, if necessary.
Answer $x=$ $\square$

Studdy Solution

STEP 1

What is this asking? We need to find the length of a side in a right triangle, knowing an angle and another side. Watch out! Make sure your calculator is in degree mode, not radians!
Also, remember SOH CAH TOA.

STEP 2

1. Set up the trigonometric ratio
2. Solve for $x$
3. Round to the nearest tenth

STEP 3

Alright, we've got a right triangle, a known angle, and we're looking for the length of the side opposite to the angle.
The other side we know is adjacent to the angle.
Opposite and adjacent?
That screams tangent!
Remember SOH CAH TOA?
TOA stands for Tangent is Opposite over Adjacent.

STEP 4

So, we can set up our equation like this: $\tan(61^\circ) = \frac{x}{11}$ Remember, $\tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}}$ .
Here, the angle is 61 degrees, the opposite side is $x$ , and the adjacent side is 11.

STEP 5

To get $x$ by itself, we need to multiply both sides of the equation by 11.
We're doing this to divide the right side by one, since $11 \cdot \frac{x}{11} = \frac{11}{11} \cdot x = 1 \cdot x = x$ . $11 \cdot \tan(61^\circ) = 11 \cdot \frac{x}{11}$

STEP 6

This simplifies to: $x = 11 \cdot \tan(61^\circ)$

STEP 7

Now, grab your calculator, make sure it's in degree mode, and calculate the value! $x \approx 11 \cdot 1.804047755$ $x \approx 19.84452531$

STEP 8

The problem asks us to round to the nearest tenth.
Looking at our result, the tenths place is 8, and the next digit is 4.
Since 4 is less than 5, we round down.

STEP 9

So, our final answer is approximately $x \approx 19.8$ .

STEP 10

$x \approx 19.8$

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