Studdy AI Logo
Math  /  Trigonometry

QuestionGiven the triangle below, find sin⁡θ\sin \theta and cos⁡θ\cos \theta.
Round your answers to three decimal places. (a) sin⁡θ=\sin \theta= i □\square ! (b) cos⁡θ=\cos \theta= i □\square □\square !

Studdy Solution

STEP 1

What is this asking? We need to find the sine and cosine of an angle θ\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⁡(θ)\sin(\theta).

STEP 4

We're given that the **opposite side** to θ\theta has length **45**, and the **hypotenuse** has length **51**.
So, we can **write** sin⁡(θ)\sin(\theta) as a fraction: sin⁡(θ)=OppositeHypotenuse=4551 \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: 4551≈0.88235 \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⁡(θ)\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⁡(θ)\cos(\theta).

STEP 8

We're given that the side **adjacent** to θ\theta has length **24**, and the **hypotenuse** has length **51**.
So, we can **write** cos⁡(θ)\cos(\theta) as a fraction: cos⁡(θ)=AdjacentHypotenuse=2451 \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: 2451≈0.47058 \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⁡(θ)\cos(\theta) is **0.471**.

STEP 11

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

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord