How to tell how many answers there are? (Trig equations)

[Tyrion101]

I'm thinking I may not understand what is a possible solution of a problem and what not. The interval for all of my problems is 0 to 2pi. So I suppose that if my answer was Sec=2, that I'd convert it to Cos, then it would be all of the angles that x = 1/2. I think I have this much right, my confusion is when there are two answers, such as the one listed above, and say sec = -1. What do I do here?

 

[Mark44]

I'm thinking I may not understand what is a possible solution of a problem and what not.

A solution of an equation is a number that makes the equation a true statement. For example, the equation x² - 3x + 2 = 0 is true only for x = 1 or x = 2. Any other value of x gives a value on the left side different from zero.

The interval for all of my problems is 0 to 2pi. So I suppose that if my answer was Sec=2, that I'd convert it to Cos, then it would be all of the angles that x = 1/2.

Yes, that's right.

I think I have this much right, my confusion is when there are two answers, such as the one listed above, and say sec = -1. What do I do here?

Let's look at the equation sec(x) = 2 first before going off to another problem. Within the interval [0, $2\pi$], how many numbers are there for which cos(x) = 1/2? Looking at the unit circle is very helpful.

If the equation were sec(x) = -1, then equivalently, cos(x) = -1. For this equation, there is only one solution. Again, the unit circle is helpful.