Find cosθ, in degrees, 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.
To find the value of cosθ in degrees, we need to use the given information that the y-coordinate of the point where the terminal side of θ intersects the unit circle is √2/2.
In the first quadrant, where θ is located, both the cosine (cos) and sine (sin) functions are positive. Since the y-coordinate is √2/2, we can determine that sinθ = √2/2.
Using the Pythagorean identity sin²θ + cos²θ = 1, we can find cosθ:
sin²θ + cos²θ = 1
(√2/2)² + cos²θ = 1
(2/4) + cos²θ = 1
1/2 + cos²θ = 1
cos²θ = 1 - 1/2
cos²θ = 1/2
To find cosθ, we take the square root of cos²θ:
cosθ = √(1/2)
Since θ is in the first quadrant, the value of cosθ is positive. Therefore:
cosθ = √(1/2) = √2/√2 = 1/√2 = √2/2.
Hence, cosθ in degrees is √2/2.