Ana was asked to use triangles formed by angles on the unit circle to prove the Pythagorean Identity sin2θ+cos2θ=1, where sinθ=1617 and π2<θ<π. What value should she use for cosθ?(1 point)
Responses
117√
Start Fraction 1 over Start Root 17 End Root End Fraction
−117√
negative Start Fraction 1 over Start Root 17 End Root End Fraction
−33√17
negative Start Fraction Start Root 33 End Root over 17 End Fraction
33√17
To prove the Pythagorean Identity sin^2θ + cos^2θ = 1, Ana can use the given value sinθ = 16/17 and the fact that θ is in the third quadrant of the unit circle (π/2 < θ < π). In the third quadrant, sinθ is negative and cosθ is positive.
Since sinθ = 16/17 and sinθ is negative in the third quadrant, Ana can use -16/17 for sinθ.
Now, using the Pythagorean Identity, we have:
sin^2θ + cos^2θ = 1
(-16/17)^2 + cos^2θ = 1
256/289 + cos^2θ = 1
To solve for cosθ, we subtract 256/289 from both sides:
cos^2θ = 1 - 256/289
cos^2θ = 289/289 - 256/289
cos^2θ = 33/289
To find cosθ, we take the square root of both sides:
cosθ = ±√(33/289)
Since cosθ is positive in the third quadrant, the value for cosθ is:
cosθ = √(33/289) = √33/17
Therefore, the correct value for cosθ is 33√17/17.