Question

Given a unit circle, what is the value of y in the first quadrant corresponding to an x-coordinate of 5/8?

Answer 1

`sqrt39/8` or `~~0.78`

Explanation:

In a unit circle, the `x` coordinate represents the `cos` value, and the `y` coordinate represents the `sin` value. Thus, we can say

`cosx=5/8`

From SOH-CAH-TOA, we know that this means if we have a right triangle, the adjacent side is `5` and the hypotenuse is `8`.

We can find the `sin` value using the Pythagorean Theorem

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

We can plug our values in to get

`5^2+b^2=8^2`

`=>b^2+25=64`

`=>b^2=39=>b=sqrt39`

We know that sine is equal to opposite over the hypotenuse. We essentially just found the opposite side, so we can plug in to get

`sinx=sqrt39/8`, which is approximately `0.78`.

This is the `y` coordinate.

Hope this helps!

Answer 2

`y = 0.78`

Explanation:

Picture a right triangle with corners made up of the point on the circle (whose x coordinate is 5/8) and the projections of that point on the x and y axes. The point on the x axis is 5/8 of the way from the origin to the circle. Since this is a unit circle, the hypotenuse by definition = 1.

Using Pythagoras:
`x^2 + y^2 = h^2`
where x and y are the coordinates of the spot on the unit circle and h is the length of the hypotenuse which, as I said, is 1.

`(5/8)^2 + y^2 = 1^2`

`0.625^2 + y^2 = 0.391 + y^2 = 1`

`y = sqrt(1-.391) = 0.78`

Another approach:
Let the angle between a line from the origin to the spot on the unit circle be `theta`. The x and y coordinates are

`x = 1*costheta and y = 1*sintheta`.

It is given that x = 5/8.

`theta = cos^-1(5/8) = 51.3^@`

`y = 1*sintheta = 1*sin51.3^@ = 0.78`

I hope this helps,
Steve