Question

What are the remaining trig functions? and how do i solve for them? pls help

Answer 1

You're told that tan(θ) is positive, but

tan(θ) = sin(θ)/cos(θ)

and you're also told that sec(θ) = 1/cos(θ) = -3. So if cos(θ) is negative, sin(θ) must also be negative. In turn, both sec(θ) = 1/cos(θ) and csc(θ) = 1/sin(θ) are also negative.

Now, recall the Pythagorean identity,

cos²(θ) + sin²(θ) = 1

Multiply through both sides by 1/cos²(θ) to get an alternate form of the identity,

1 + tan²(θ) = sec²(θ)

Solve for tan(θ) (which we know is positive):

tan(θ) = √(sec²(θ) - 1) = 2√2

Right away, we get

cot(θ) = 1/tan²(θ) = 1/(2√2) = √2/4

Since sec(θ) = -3, it follows that cos(θ) = -1/3.

Then

tan(θ) = sin(θ)/cos(θ)   ==>   sin(θ) = 2√2 × (-1/3) = -2√2/3

and so

csc(θ) = 1/sin(θ) = -3/(2√2)