Use the Pythagorean Identity and Tangent Identity to find tanθ if sinθ=-1 and 0≤θ<2π radians. (3 points)
HINT: Find all tanθ on the unit circle when sinθ=-1
Since sinθ=-1, we know that θ is in the fourth quadrant of the unit circle where the y-coordinate is -1. In this case, the x-coordinate is √3, so the point on the unit circle is (√3,-1).
Using the Pythagorean Identity, we have:
sin^2θ + cos^2θ = 1
(-1)^2 + (√3)^2 = 1
1 + 3 = 1
4 = 1
Therefore, cosθ = ±√(1-(-1)^2) = ±√(1-1) = ±√0 = 0
Since cosθ is 0, tanθ = sinθ/cosθ = (-1)/0 = undefined.
Therefore, tanθ is undefined when sinθ = -1.