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Math  /  Trigonometry

QuestionFind sin⁡θ\boldsymbol{\operatorname { s i n }} \theta. csc⁡θ=−83sin⁡θ=□\begin{array}{l} \csc \theta=-\frac{8}{3} \\ \sin \theta=\square \end{array} (Simplify your answer, including any radicals. Use integers or fractions for any

Studdy Solution

STEP 1

What is this asking? We're given the cosecant of an angle and need to find the sine of that same angle. Watch out! Don't mix up sine and cosine!
Remember the relationship between sine and cosecant.

STEP 2

1. Relate Sine and Cosecant
2. Calculate Sine

STEP 3

Alright, let's **kick things off**!
We know that cosecant is the reciprocal of sine.
Think of it like this: if sine is how tall something is, cosecant is how long something is when you flip it over!
Mathematically, this means: csc⁡θ=1sin⁡θ \csc \theta = \frac{1}{\sin \theta}

STEP 4

We're given that csc⁡θ=−83\csc \theta = -\frac{8}{3}.
Let's **plug this in** to our equation: −83=1sin⁡θ -\frac{8}{3} = \frac{1}{\sin \theta}

STEP 5

Now, we want to **solve for** sin⁡θ\sin \theta.
It's like having sin⁡θ\sin \theta trapped in the denominator, and we need to set it free!
To do this, we can **multiply both sides** of the equation by sin⁡θ\sin \theta: −83⋅sin⁡θ=1 -\frac{8}{3} \cdot \sin \theta = 1 Remember, we're multiplying both sides by sin⁡θ\sin \theta so that we can eventually isolate it.

STEP 6

Almost there!
To **completely isolate** sin⁡θ\sin \theta, we need to **multiply both sides** by the reciprocal of −83-\frac{8}{3}, which is −38-\frac{3}{8}.
This will give us: −38⋅−83⋅sin⁡θ=1⋅−38 -\frac{3}{8} \cdot -\frac{8}{3} \cdot \sin \theta = 1 \cdot -\frac{3}{8}

STEP 7

Notice that −38⋅−83=2424=1-\frac{3}{8} \cdot -\frac{8}{3} = \frac{24}{24} = 1.
So, we're left with: sin⁡θ=−38 \sin \theta = -\frac{3}{8} Boom! We've found sin⁡θ\sin \theta!

STEP 8

sin⁡θ=−38\sin \theta = -\frac{3}{8}.

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