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

Algebra ->  Trigonometry-basics -> 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      Log On


   

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) About Me  (Show Source):
You can put this solution on YOUR website!

theta = theta = Greek letter often used with angles.

The restriction on theta is that -pi%2F2+%3C+theta+%3C+pi%2F2 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 cos%28theta%29+%3E+0 since the x coordinates of (x,y) are positive.
We are to the right of the y axis.
Recall that x+=+cos%28theta%29 for the point (x,y) on the unit circle.

Let's square both sides of the first given equation
x+=+8%2Asin%28theta%29

x%5E2+=+%288%2Asin%28theta%29%29%5E2

x%5E2+=+8%5E2%2Asin%5E2%28theta%29

x%5E2+=+64%2Asin%5E2%28theta%29

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.
x%2F%28sqrt%2864-x%5E2%29%29

%288%2Asin%28theta%29%29%2F%28sqrt%2864-64%2Asin%5E2%28theta%29%29%29

%288%2Asin%28theta%29%29%2F%28sqrt%2864%281-sin%5E2%28theta%29%29%29%29 Factor out the GCF 64.

%288%2Asin%28theta%29%29%2F%28sqrt%2864%2Acos%5E2%28theta%29%29%29 Use the identity cos^2 = 1 - sin^2

%288%2Asin%28theta%29%29%2F%28sqrt%2864%29%2Asqrt%28cos%5E2%28theta%29%29%29 Break up the square root in the denominator

%288%2Asin%28theta%29%29%2F%288%2Acos%28theta%29%29 Simplify each square root. Cosine is positive in quadrants Q1 and Q4.

%28sin%28theta%29%29%2F%28cos%28theta%29%29

tan%28theta%29 Use the identity of tan = sin/cos

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

Therefore,
x%2F%28sqrt%2864-x%5E2%29%29
simplifies to
tan%28theta%29
when x+=+8%2Asin%28theta%29 such that -pi%2F2+%3C+theta+%3C+pi%2F2

In other words,
x%2F%28sqrt%2864-x%5E2%29%29=tan%28theta%29
is an identity when x+=+8%2Asin%28theta%29 such that -pi%2F2+%3C+theta+%3C+pi%2F2

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