Given sine, theta, equals, minus, start fraction, 4, divided by, 7, end fractionsinθ=−
7
4
and angle thetaθ is in Quadrant IV, what is the exact value of cosine, thetacosθ in simplest form? Simplify all radicals if needed.
Answer
Attempt 1 out of 2
cosine, theta, equalscosθ=
We know that sine of an angle is equal to the ratio of the length of the side opposite the angle to the length of the hypotenuse. In this case, we have sine θ = -7/4, which means the opposite side has a length of -7 and the hypotenuse has a length of 4.
Since angle theta is in Quadrant IV, we can draw a right triangle in Quadrant IV such that the adjacent side is positive. Let's call the adjacent side x.
Using the Pythagorean theorem, we can find the length of the adjacent side:
x^2 + (-7)^2 = 4^2
x^2 + 49 = 16
x^2 = 16 - 49
x^2 = -33
Since x must be positive, we can conclude that x does not have a real value. Therefore, cosine θ is undefined in this case.