Given cotθ = -3 and cosθ < 0:
sinθ = -√(1 - cos²θ)
cosθ < 0
tanθ = -1/3
cotθ = -3
secθ = -1/cosθ
cscθ = -1/sinθ
We are given that cot(θ) = -3 and cos(θ) < 0. Let's evaluate the six trigonometric functions based on this information.
Sine (sinθ):
We know that
cot(θ) = 1/tan(θ), and tan(θ) = 1/cot(θ).
Therefore, tan(θ) = -1/3.
Using the Pythagorean identity sin²θ + cos²θ = 1,
We find sinθ as √(1 - cos²θ).
Given that cos(θ) < 0, we take the negative square root: sinθ = -√(1 - cos²θ).
Cosine (cosθ):
We are given cos(θ) < 0.
Tangent (tanθ):
tan(θ) = -1/3.
Cotangent (cotθ):
cot(θ) = -3 (given).
Secant (secθ):
sec(θ) is the reciprocal of cos(θ). Since cos(θ) < 0, sec(θ) = -1/cos(θ).
Cosecant (cscθ):
csc(θ) is the reciprocal of sin(θ). Therefore, cscθ = -1/sinθ.