Question

If cos 1/3 and tan is angle < 0 find the exact value

Answer 1

Since tanθ < 0 and cosθ > 0, we see that sinθ < 0.

We have:

cos²θ + sin²θ = 1
==> sinθ = -√(1 - cos²θ) = -√[1 - (1/3)²] = -√(8/9) = -2√2/3.

Thus:

sinθ = -2√2/3
cosθ = 1/3
secθ = 1/cosθ = 1/(1/3) = 3
cscθ = 1/sinθ = 1/(-2√2/3) = -3√2/4
tanθ = sinθ/cosθ = (-2√2/3)/(1/3) = -2√2
cotθ = 1/tanθ = 1/(-2√2) = -√2/4.

Finally, θ_R = arccos(1/3) ≈ 71° and:

θ = 180° - 71° = 109°.

I hope this helps!