Ana was asked to use triangles formed by angles on the unit circle to prove the Pythagorean Identity sin^2 θ +cos^2 θ = 1, where sin θ = 16/17 and π/2 < θ < π What value should she use for cos θ?
The Pythagorean Identity for trigonometry states that:
sin^2(θ) + cos^2(θ) = 1
Ana was given that sin(θ) = 16/17 and is asked to prove the Pythagorean Identity using this information, where θ is an angle in the second quadrant (since π/2 < θ < π).
In the unit circle, the sine of an angle θ is equal to the y-coordinate of the point on the unit circle corresponding to that angle, and the cosine of an angle is equal to the x-coordinate of that same point. Because we know that the radius of the unit circle is 1, and we have sin(θ) = y/r (where r is the radius), for the unit circle where r = 1, sin(θ) just equals y.
Given that sin(θ) = 16/17, we can say that the y-coordinate of the point on the unit circle at angle θ is 16/17.
Using the Pythagorean Identity, we want to find cos(θ), which correlates to the x-coordinate on the unit circle:
sin^2(θ) + cos^2(θ) = 1
(16/17)^2 + cos^2(θ) = 1
256/289 + cos^2(θ) = 1
cos^2(θ) = 1 - 256/289
cos^2(θ) = 289/289 - 256/289
cos^2(θ) = (289 - 256) / 289
cos^2(θ) = 33/289
Now we take the square root of both sides to solve for cos(θ):
cos(θ) = ±√(33/289)
We need to determine whether to use the positive or negative square root. Since θ is in the second quadrant (π/2 < θ < π), the cosine of θ (which corresponds to the x-coordinate on the unit circle) is negative, because the x-coordinate is to the left of the y-axis in the second quadrant.
Therefore:
cos(θ) = -√(33/289)
cos(θ) = -√(33)/√(289)
cos(θ) = -√(33)/17
Ana should use -√(33)/17 for cos(θ) to prove the Pythagorean Identity given that sin(θ) = 16/17 and θ is in the second quadrant.