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Math  /  Trigonometry

Question15 cm H 112 24° x adj OP MAT

Studdy Solution

STEP 1

What is this asking? We've got a right triangle, know the angle and the opposite side, and need to find the adjacent side! Watch out! Make sure your calculator is in degree mode, not radians!

STEP 2

1. Tangent Relationship
2. Solve for *x*

STEP 3

Alright, so we've got a right triangle here, which means we can use our trigonometric friends!
We know the angle, which is **24°**, and the opposite side, which is **15 cm**.
We're looking for the adjacent side, which we'll call *x*.
Which trig function relates the opposite and adjacent sides of a right triangle?
That's right, it's the **tangent** function!

STEP 4

Let's remember what the tangent function tells us: tan⁡(θ)=oppositeadjacent \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} In our case, θ\theta is our **24°** angle, the opposite side is **15 cm**, and the adjacent side is our mysterious *x*.
Let's plug those values in: tan⁡(24∘)=15 cmx \tan(24^\circ) = \frac{15 \text{ cm}}{x}

STEP 5

Now, we want to get *x* all by itself.
To do that, we can multiply both sides of the equation by *x*: x⋅tan⁡(24∘)=x⋅15 cmx x \cdot \tan(24^\circ) = x \cdot \frac{15 \text{ cm}}{x} On the right side, the *x* in the numerator and the *x* in the denominator divide to one, leaving us with: x⋅tan⁡(24∘)=15 cm x \cdot \tan(24^\circ) = 15 \text{ cm}

STEP 6

Now, we just need to divide both sides by tan⁡(24∘)\tan(24^\circ) to isolate *x*: x=15 cmtan⁡(24∘) x = \frac{15 \text{ cm}}{\tan(24^\circ)}

STEP 7

Time to grab our calculators!
Make sure it's in degree mode.
Now, calculate tan⁡(24∘)\tan(24^\circ), which is approximately **0.445**.
So, we have: x=15 cm0.445 x = \frac{15 \text{ cm}}{0.445}

STEP 8

Finally, divide **15 cm** by **0.445** to find the value of *x*: x≈33.7 cm x \approx 33.7 \text{ cm}

STEP 9

The length of the adjacent side, *x*, is approximately **33.7 cm**.

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