All right. Let's look at another example.

Again, let's solve this equation only on the interval zero to two pi.

We can begin by subtracting two from both sides,

which gives us secant of theta is equal to negative 2.

Now, remember that secant of theta is equal to 1 divided by cosine of theta,

or cosine of theta is equal to 1 divided by secant of theta.

And, therefore, if secant of theta is equal to negative two,

then we're looking for angles theta

such that cosine of theta is one divided by negative 2,

or negative one half.

Again, recalling our units circle,

and remembering that the cosine of theta is the x coordinate

of the point of intersection of the terminal side of the angle and the unit circle.

We see that the x coordinate of this point here in quadrant two is negative one half,

which corresponds to the angle of 2 pi divided by 3.

But, also down here in Quadrant three.

The x coordinate of this point is also negative one half,

which corresponds to the angle of 4 pi divided by 3.

Therefore, our answer here would be theta is equal to,

2 pi divided by 3,

or 4 pi divided by 3.

All right.

Let's look at one more example.

Again, let's find all solutions to this equation but only in the interval zero to two pi.

We'll begin by adding one to both sides,

which gives us cotangent of theta,

is equal to 1.

Now, remember that cotangent of theta,

is equal to cosine of theta,

divided by sine of theta.

But, we just saw that the cosine of theta was the x coordinate of

the point of intersection of the terminal side of the angle and the unit circle,

and the sine of theta was the y coordinate.

So, this would be equal to x divided by y.

So, therefore, we are looking for angles in the interval zero to two pi,

where this ratio of x to y is equal to 1.

And, this will happen at pi over 4,

as well as 5 pi over 4,

because remember our unit circle,

at pi over 4,

the ratio of x to y here will be equal to 1,

but also down here in Quadrant three,

the ratio of x to y will also equal 1,

which corresponds to the angle of 5 pi divided by 4.

And don't forget this angle down here in Quadrant three,

most students will only remember the angle in Quadrant one.

But, the ratio of x to y in Quadrant three will also yield a positive number.

Therefore, our answer here is,

theta is equal to pi divided by 4 or 5 pi divided by 4.