The point P is on the unit circle. If the y-coordinate of P is −3/5, and P is in quadrant III , then

Question

The point  is on the unit circle. If the y-coordinate of  is , and  is in quadrant III , then

 =   

Answer 1

If you recall from trigonometry, the unit circle is a special circle centered on the origin of a coordinate grid, with a radius equal to 1 unit.

Since we know P is on the unit circle, we know that the length of the line from P to the origin is exactly 1 unit.

We're also given a y-coordinate, y = -3/5. Note how the y-coordinate expresses the length of a perpendicular line from point P to the x-axis.

We know the length from P to the origin, and the perpendicular length from P to the x-axis. Using this information, we can construct a right triangle.

And with a right triangle, we can use Pythagorean theorem.

If x is the x-coordinate, y is the y-coordinate, and h is the hypotenuse length, we can set up the equation:

x² + y² = h²

We can then solve for x:

x² = h² – y²

x = √h² – y²

We can now substitute the information we know:

h = 1 and y = -3/5, so

x = √h² – y²

x = √(1)² – (-3/5)²

x = √1 – 9/25

x = √16/25

x = 4/5

We now have a value for x. But, we need to know if it is positive or negative. The question noted that P is in quadrant III, which has negative values for both x and y. So, x is negative.

Thus, x = –4/5