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Math  /  Geometry

QuestionFind DFD F.
Write your answer as an integer or as a decimal rounded to the nearest tenth.

Studdy Solution

STEP 1

What is this asking? We're looking for the length of one side of a right triangle when we know the hypotenuse and an angle. Watch out! Make sure your calculator is in degree mode, not radians!

STEP 2

1. Set up the cosine ratio.
2. Solve for the unknown side.

STEP 3

Alright, let's **dive in**!
We've got a right triangle, a known angle, and we're looking for the side *adjacent* to that angle.
This screams "**cosine**"!
Remember, cosine is all about the adjacent side and the hypotenuse.

STEP 4

The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.
So, we can write: cos⁡(∠D)=adjacent sidehypotenuse \cos(\angle D) = \frac{\text{adjacent side}}{\text{hypotenuse}}

STEP 5

Let's **plug in** what we know!
Our angle ∠D\angle D is **61 degrees**, the adjacent side is DFDF, which is what we're trying to find, and the hypotenuse DEDE is **5.8**.
This gives us: cos⁡(61∘)=DF5.8 \cos(61^\circ) = \frac{DF}{5.8}

STEP 6

Now, we want to get DFDF all by itself.
To do this, we'll **multiply both sides** of the equation by **5.8**.
Remember, what we do to one side, we *must* do to the other to keep things balanced!
This gives us: 5.8⋅cos⁡(61∘)=5.8⋅DF5.8 5.8 \cdot \cos(61^\circ) = 5.8 \cdot \frac{DF}{5.8}

STEP 7

On the right side, the **5.8's divide to one**, leaving us with just DFDF: 5.8⋅cos⁡(61∘)=DF 5.8 \cdot \cos(61^\circ) = DF

STEP 8

Now, grab your calculator and make sure it's in **degree mode**!
Calculate the value of 5.8⋅cos⁡(61∘)5.8 \cdot \cos(61^\circ): DF=5.8⋅cos⁡(61∘)≈5.8⋅0.4848≈2.808 DF = 5.8 \cdot \cos(61^\circ) \approx 5.8 \cdot 0.4848 \approx 2.808

STEP 9

Rounding to the nearest tenth gives us our **final answer**: DF≈2.8 DF \approx 2.8

STEP 10

DFDF is approximately **2.8**.

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