Answer:
tan(θ) is undefined.
Step-by-step explanation:
Recall that tan(θ) = sin(θ) / cos(θ). We are given that sin(θ) = -1. Hence:
[tex]\displaystyle \tan\theta=\frac{\sin\theta}{\cos\theta}=-\frac{1}{\cos\theta}[/tex]
Since 1 / cos(θ) = sec(θ):
[tex]\tan\theta=-\sec\theta[/tex]
We can square both sides:
[tex]\tan^2\theta=\sec^2\theta[/tex]
From the Pythagorean Identity:
[tex]\tan^2\theta+1=\sec^2\theta[/tex]
Substitute:
[tex]\tan^2\theta+1=\tan^2\theta[/tex]
So:
[tex]1\neq0[/tex]
Since we acquire an untrue statement, no solutions exist.
If needed, refer to the unit circle. Recall that sin(θ) only equals -1 when θ = 3π/2 (on the interval [0, 2π)). At 3π/2, cos(θ) = 0. Since tan(θ) = sin(θ) / cos(θ), tan(θ) is undefined whenever sin(θ) = -1 (or 1).
