SOLUTION: Let x = 8 sin(𝜃), − 𝜋/2 < 𝜃 < 𝜋/2. Simplify the expression. x ----------- - square together 64 − x2 I know the answer is tan(𝜃) but I don't r

Question 1199158: Let
x = 8 sin(𝜃), − 𝜋/2 < 𝜃 < 𝜋/2.
Simplify the expression.

x
-------------
square together 64 − x2

I know the answer is tan(𝜃) but I don't remember how i got the answer i lost the paper im sorry
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

theta = = Greek letter often used with angles.

The restriction on theta is that which places theta either

(a) in quadrant Q1 in the northeast
(b) on the positive x axis
(c) in quadrant Q4 in the southeast

No matter where theta is located, it makes since the x coordinates of (x,y) are positive.
We are to the right of the y axis.
Recall that for the point (x,y) on the unit circle.

Let's square both sides of the first given equation

Here is a list of trig identities
https://tutorial.math.lamar.edu/pdf/Trig_Cheat_Sheet.pdf
Two specific identities we need are:

cos^2 = 1 - sin^2
tan = sin/cos

That notation is a bit informal, but I hope you can get the idea.

Now let's plug in those x and x^2 items into the expression below.
From there we simplify.

Factor out the GCF 64.

Use the identity cos^2 = 1 - sin^2

Break up the square root in the denominator

Simplify each square root. Cosine is positive in quadrants Q1 and Q4.

Use the identity of tan = sin/cos

-----------------------------------------------

Therefore,

simplifies to

when such that

In other words,

is an identity when such that

I recommend using graphing software such as GeoGebra or Desmos to visually confirm the answer.

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