Question

Fill In The Blanks. PLEASE HELP.
sin−1 x + sin−1(−x) = 0
Let α = sin−1 x and β = sin−1(−x)
. These equations imply that
sin α = x
sin β = -x
Looking at the right triangles for these values,
cos α = ________
cos β = ________
Making these substitutions, we see that
sin−1 x + sin−1(−x) = α + β
= sin−1[sin(α + β)]
= sin−1(sin α cos β + sinβ cosα
= sin−1 · (x√(1-x^2) + _________
= sin−1 · ____________
= __________

Answer 1

\[\Large \sin\alpha=\frac{opposite}{hypotenuse} \qquad\to\qquad \sin \alpha=\frac{x}{1}\]

Answer 2

Understand how I set up that triangle?
I rewrote x and x/1. Allowing me to label the corresponding sides.
We need to solve for the missing side using the Pythagorean Theorem.

Answer 3

it's actually sin^-1(-x) my bad

Answer 4

We're dealing with the first term.\[\Large \sin^{-1}(x)=\alpha \qquad\to\qquad \sin \alpha=x\]
We can do the other one after this :o

Answer 5

howbout cosα then?

Answer 6

We have to setup a triangle for alpha, then another triangle for beta.
Do you understand how I setup the triangle?
We can't find cos alpha until we figure out that missing side along the bottom.

Answer 7

so it would be √(1-x^2) am i right?

Answer 8

ya that sounds right :)