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

2 Answers
Aug 2, 2018

Answer:

#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!

Aug 3, 2018

Answer:

#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