Find cosθ
if the y-coordinate of the point where the terminal side of θ
intersects the unit circle is 2√2
, given that θ
is an acute angle and in the first quadrant.(1 point)
cosθ=
.
Since θ is an acute angle in the first quadrant, the x-coordinate of the point where the terminal side of θ intersects the unit circle is positive.
Let's call the x-coordinate of the point P. By the definition of the unit circle, we know that the x-coordinate of any point on the unit circle is equal to the cosine of the angle formed with the positive x-axis.
Therefore, cosθ = x-coordinate of point P.
Since we are given that the y-coordinate of point P is 2√2, we can use the Pythagorean theorem to find the length of the hypotenuse formed by the point P, the origin, and the positive x-axis.
Using the formula: hypotenuse² = x-coordinate² + y-coordinate², we can plug in the values to find:
(hypotenuse)² = (x-coordinate)² + (2√2)²
= (x-coordinate)² + 4(2)
= (x-coordinate)² + 8
Since the point P lies on the unit circle, the length of the hypotenuse is 1. So we have:
1² = (x-coordinate)² + 8
1 = (x-coordinate)² + 8
(x-coordinate)² = 1 - 8
(x-coordinate)² = -7
Since θ is an acute angle in the first quadrant, which means it is between 0 and 90 degrees, the x-coordinate must be positive. Therefore, we can conclude that there is no solution for cosθ in this scenario.
cosθ = N/A