Math Snap

STEP 1

What is this asking?
We need to find the sine and cosine of an angle $\theta$ in a right triangle, given the lengths of all three sides.
Watch out!
Remember SOH CAH TOA!
Don't mix up which side is opposite, adjacent, or the hypotenuse.

STEP 2

1. Find the sine of theta.
2. Find the cosine of theta.

STEP 3

Alright, let's dive in!
Remember SOH CAH TOA?
Sine is Opposite over Hypotenuse!
We're looking for $\sin(\theta)$.

STEP 4

We're given that the opposite side to $\theta$ has length 45, and the hypotenuse has length 51.
So, we can write $\sin(\theta)$ as a fraction:
$$\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{45}{51}$$

STEP 5

Now, let's convert this fraction to a decimal by dividing 45 by 51:
$$\frac{45}{51} \approx 0.88235$$

STEP 6

The problem asks us to round to three decimal places.
Looking at the fourth decimal place, which is 3, we see it's less than 5, so we round down.
Our final answer for $\sin(\theta)$ is 0.882.

STEP 7

Now, let's tackle the cosine!
Remember SOH CAH TOA?
Cosine is Adjacent over Hypotenuse!
We're looking for $\cos(\theta)$.

STEP 8

We're given that the side adjacent to $\theta$ has length 24, and the hypotenuse has length 51.
So, we can write $\cos(\theta)$ as a fraction:
$$\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{24}{51}$$

STEP 9

Let's convert this fraction to a decimal by dividing 24 by 51:
$$\frac{24}{51} \approx 0.47058$$

STEP 10

Again, we need to round to three decimal places.
Looking at the fourth decimal place, which is 5, we round up.
Our final answer for $\cos(\theta)$ is 0.471.

We found that $\sin(\theta) = 0.882$ and $\cos(\theta) = 0.471$.