What are possible solutions to equation `2cos(theta)-1=sec(theta)`?

Expert Answers
txmedteach eNotes educator| Certified Educator

Every time you see questions like this in trigonometry, I want you to start first by trying to get every trigonometric function into forms of sine and cosine of the same angle. This sort of problem is solved almost every time by performing this sort of conversion and solving algebraically!

In our case, we are given the following:

`2costheta - 1 = sectheta`

Recall that `sectheta = 1/costheta`, so we can make the following conversion on the right side of the equation:

`2costheta-1= 1/costheta`

Now, let's solve this algebraically, as if `costheta` were simply "`x`." Start by multiplying both sides by `costheta`:

`2cos^2theta-costheta = 1`

Now, subtract 1 from both sides:

`2cos^2theta - costheta -1 = 0`

We can now factor this equation in the following way:

`(2costheta + 1)(costheta-1) = 0`

Now, we have a situation you may be familiar with. We must set both expressions to zero to get the complete solution. Let's start with the first expression:

`2costheta + 1 = 0`

Solve for `costheta` by subtracting 1, then dividing by 2:

`costheta = -1/2`

Now, you just need to recall when `costheta = -1/2`. You might consult a trigonometry table, or just remember that this occurs when we are at angles of `2/3 pi` and `4/3 pi`:

`theta = 2/3 pi` or `theta = 4/3 pi`

However, we need to find all of the answers and cosine is a periodic function with period `2pi`! Therefore, we know that for any integer `k_n` that the following is the set of all solutions:

`theta = 2/3 pi + 2pik_1` or `theta = 4/3pi+2pik_2`

Now, we need to solve for the second expression in our factored equation:

`costheta -1 = 0`

Solve this expression by first adding 1 to both sides:

`costheta = 1`

There is only one spot on our unit circle where `costheta = 1`, thankfully, and it is given by the following solution:

`theta = 0`

Again, cosine is a periodic function with period 2pi, so we know that the following is the full set of solutions given by this value for `theta`:

`theta = 2pik_3`

We now have all possible solutions for `theta`:

`theta = 2/3pi + 2pik_1` or `theta = 4/3pi+2pik_2` or `theta = 2pik`

You might notice a pattern in the above 3 solutions. Each is a multiple of `2/3 pi`! Therefore, we can simplify our solution to the following result:

`theta = 2/3pik`

If you don't see how this solution encompasses all three sets of solutions above, try to convince yourself! I hope this helps!