Given cosine, theta, equals, start fraction, 4, divided by, 7, end fractioncosθ=
7
4
and angle thetaθ is in Quadrant I, what is the exact value of sine, thetasinθ in simplest form? Simplify all radicals if needed.
Answer
Attempt 1 out of 2
sine, theta, equalssinθ=
The given equation is cos(θ) = 4/7.
Since the angle θ is in the first quadrant, both the sine and cosine of θ will be positive.
To find sin(θ), we can use the Pythagorean identity: sin^2(θ) + cos^2(θ) = 1.
Plugging in the given value of cos(θ) = 4/7, we have:
sin^2(θ) + (4/7)^2 = 1
sin^2(θ) + 16/49 = 1
sin^2(θ) = 1 - 16/49
sin^2(θ) = 49/49 - 16/49
sin^2(θ) = 33/49
Taking the square root of both sides, we have:
sin(θ) = √(33/49)
Now, since the angle θ is in the first quadrant, the sine value will be positive. Therefore:
sin(θ) = √(33/49) = √33/7.
So the exact value of sin(θ) is √33/7.