Need help on these problems...

Question

Hi, I am having some difficulty with this problem. I am given that cos θ= -2/5 , and that π/2 < θ < π.

I then need to use this information to evaluate cos 2θ and tan θ/2.

I drew a triangle, and got θ=113.578, which I don't think is right.

How can I solve this? Thank you in advance.

Answer 1

cos2θ = 2cos²θ - 1 = 2(-2/5)² - 1 = 8/25 - 1 = - 17/25;

tanθ/2 = √(1 - cosθ)/(1 + cosθ) = √(1 + 2/5)/(1 - 2/5) = √7/3 = √21/3, because θ/2 in 1st quadrant

Answer 2

First, the good news! You're actually correct about what theta equals. If you calculate the arc cosine of -2/5, then you will get θ=113.578. This also makes sense with the domain given in the problem (π/2 < θ < π), which is in the second quadrant.

However, I'm going to guess that your teacher may be looking for exact values, so let's see how we can get those values.

Let's start with cos 2θ.

The identity that we can use is: cos 2θ = 2 (cos θ)^2 - 1

Since we know that cos θ= -2/5, we can substitute that value in.

So cos 2θ = 2 (-2/5)^2 - 1 = -0.68

Now, let's do tan θ/2.

There are several formulas we can use, but the easiest might be:

tan θ/2 = +/- sqrt [(1-cos θ)/(1 + cos θ)]

Again, substitute the value for cos θ.

tan θ/2 = +/- sqrt [(1+ 2/5)/(1 - 2/5)] = +/- sqrt (7/3)

Because θ/2 would be in the first quadrant, tan θ/2 will be positive.

tan θ/2 = sqrt (7/3)