Question

Find the Sin?

Answer 1

`1/3sqrt(13/3)`

Explanation:

The sine of an angle is equal to the side opposite to that angle divided by the hypotenuse.

Here, we see our hypotenuse, the side opposite to the right angle, is `3sqrt(6).`

Looking at `X`, the angle whose sine we want, the side opposite to `X` is unknown. Therefore, we'll need to calculate it using the Pythagorean Theorem:

`a^2+b^2=c^2`

Where `a, b` are the legs of the triangle and `c` is the hypotenuse.

Plugging in `c=3sqrt(6), a=2sqrt(7)` (one of our legs), we get:

`(2sqrt(7))^2+b^2=(3sqrt(6))^2`

`2^2sqrt(7)^2+b^2=3^2sqrt(6)^2` (Because `(xy)^2=x^2y^2`)

`4(7)+b^2=9(6)` (Because `sqrt(x)^2=x`)

`28+b^2=54`

`b^2=54-28=26`

Square rooting both sides to solve for `b` yields:

`b=sqrt(26)`

So, now we know the side opposite to `X` is `sqrt(26)` and the hypotenuse is `3sqrt(6).`

`sin(X)=sqrt(26)/(3sqrt(6))=1/3sqrt(26/6)=1/3sqrt(13/3)` (Because `sqrt(x)/sqrt(y)=sqrt(x/y)`

Answer 2

Use Pythagorean Theorem followed by SOHCAHTOA

Explanation:

Using the Pythagorean Theorem...

`a^2 + b^2 = c^2`

... You can find the missing side length. A and B are the bases, and C is the hypotenuse.

After getting the missing side, refer to the acronym SOHCAHTOA to find out what to put on your calculator. In your case, you'll just put the simplified fraction.

SOHCAHTOA stands for

S in:
O pposite / H ypotenuse

C os:
A djacent / H ypotenuse

T an:
O pposite / A djacent

Hope this helps :)