Math  /  Geometry

Question(D)

Studdy Solution

STEP 1

What is this asking? We need to find the length of the side opposite the **30-degree angle** in a right triangle, knowing the **hypotenuse** is **5 inches** long. Watch out! Don't mix up the sine and cosine!
Remember **SOH CAH TOA**!

STEP 2

1. Set up the trigonometric ratio.
2. Solve for *x*.

STEP 3

We're dealing with a right triangle, a **30-degree angle**, and we know the **hypotenuse**.
We want to find the side **opposite** the angle.
This screams "**sine**"!
Remember **SOH** (Sine is Opposite over Hypotenuse)!

STEP 4

Let's write down the sine of the **30-degree angle**: sin(30)=oppositehypotenuse \sin(30^\circ) = \frac{\text{opposite}}{\text{hypotenuse}}

STEP 5

We know the **hypotenuse** is **5 inches** and the **opposite side** is *x* inches.
Let's plug those values into our equation: sin(30)=x5 \sin(30^\circ) = \frac{x}{5}

STEP 6

Now, we need to **isolate** *x*.
To do this, we'll **multiply** both sides of the equation by **5**.
We're multiplying by 5 to divide the right side by 5, since 55=1\frac{5}{5}=1. 5sin(30)=5x5 5 \cdot \sin(30^\circ) = 5 \cdot \frac{x}{5}

STEP 7

This simplifies to: 5sin(30)=x 5 \cdot \sin(30^\circ) = x

STEP 8

We know that sin(30)=12\sin(30^\circ) = \frac{1}{2}.
Let's **substitute** that in: 512=x 5 \cdot \frac{1}{2} = x

STEP 9

Multiplying **5** by **one-half** gives us: x=52 x = \frac{5}{2}

STEP 10

So, x=52x = \frac{5}{2} inches, or *x* = **2.5 inches**!

STEP 11

The length of the side opposite the 30-degree angle is **2.5 inches**.

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