Please help me with this math problem!?

Question

I'm just starting Pre Cal, and I don't remember ever doing this. Maybe it's just the vestiges of summer clouding my mind..But anyway, this is the problem: Suppose tanθ=5 and π<θ<(3π/2). Find the exact value of cosθ. I also need to include a "correctly labeled sketch." Help!

Answer 1

If tanθ=5 then θ=arctan 5 and θ=78.7 or 1.4 radians

tanθ = tan(θ+pi)

so to get an equivalent angle inside the parameters π&lt;θ&lt;(3π/2)

(θ+pi)= 78.7+180=258.7degrees=

cos 258.7= cos 1.43 π = -0.2

Answer 2

Yes, you will need a sketch. Draw a right-angled triangle of the third quadrant form. Put in θ as the angle.

Now, we know tan θ = 5 so using our simple trig rules, (opposite / adjacent) we know the opposite side is 5 and the adjacent side is 1.

Now we have two lengths. Use Pythagoras to find the hypotenuse (x):

x² = 1² + 5²

x² = 1 + 25

x = √26

Now that we have the hypotenuse, use simple trig again (cos) to finally get cos θ (adjacent / hypotenuse):

cos θ = 1 / √26

However, the intervals you have given us is of the third quadrant and so, cos θ must be negative:

cos θ = -1 / √26