Given cosine, theta, equals, minus, start fraction, 6, divided by, 7, end fractioncosθ=−
7
6
and angle thetaθ is in Quadrant III, what is the exact value of sine, thetasinθ in simplest form? Simplify all radicals if needed.
In Quadrant III, cosine is negative. Since we are given that cosθ = -6/7, we can determine the value of sinθ.
Using the Pythagorean identity, we know that sinθ = √(1 - cos^2θ).
Substituting in our value for cosθ:
sinθ = √(1 - (-6/7)^2)
sinθ = √(1 - 36/49)
sinθ = √(49/49 - 36/49)
sinθ = √(13/49)
sinθ = √13/7
Therefore, the exact value of sinθ is √13/7.