Given cosine, theta, equals, one quartercosθ=
4
1
and angle thetaθ is in Quadrant I, what is the exact value of sine, thetasinθ in simplest form? Simplify all radicals if needed.
We know that the cosine of an angle is equal to the adjacent side divided by the hypotenuse. In this case, we are given that the cosine of theta is equal to 1/4, so we can draw a right triangle in quadrant I where the adjacent side is 1 and the hypotenuse is 4.
Using the Pythagorean theorem, we can find the opposite side of the triangle. Let's call it "x".
x^2 = 4^2 - 1^2
x^2 = 16 - 1
x^2 = 15
Taking the square root of both sides, we get:
x = √15
Therefore, the opposite side of the triangle is √15.
The sine of theta is equal to the opposite side divided by the hypotenuse. So, sine theta = √15/4.
Therefore, the exact value of sine theta is √15/4.