Given sine, theta, equals, minus, start fraction, 2, divided by, 5, end fractionsinθ=− 5 2 and angle thetaθ is in Quadrant III, what is the exact value of cosine, thetacosθ in simplest form? Simplify all radicals if needed.
In quadrant III, the cosine function is negative.
Since sinθ = -5/2, and sinθ = opposite/hypotenuse, we can assign the opposite side to be -5 and the hypotenuse to be 2.
Using Pythagorean theorem, we can find the adjacent side:
a² + (-5)² = 2²
a² + 25 = 4
a² = 4 - 25
a² = -21
Since we are in quadrant III, the cosine function is negative, so we take the negative square root:
a = -√(-21)
Thus, the value of cosineθ is -√(-21), or, in simplest form:
cosθ = -√21