Once we understand the trigonometric functions sine, cosine, and tangent, we are ready to learn how to use inverse trigonometric functions to find the measure of the angle the function represents. Inverse trigonometric functions, found on any standard scientific or graphing calculator, are a vital part of trigonometry and will be encountered often in Calculus.
In right triangles when we're talking about cosine, sine and tangent sometimes you're going to need to use what's known an inverse trig function. Let's look at what that means, if I asked you to find the measure of angle b, so that's the angle right up here. Well what we're going to do is we're going to say are we going to use cosine, sine or tangent? Relative to b I know the opposite side and the adjacent, opposite and adjacent is tangent, so I can say that the tangent of b is equal to the ratio of 14 to 12 the opposite to the adjacent. But now we have a problem, how do I find out what b is? I know eventually I want to see b equals. Right now we have tangent of b if I went back to Algebra I would say divide both sides by tangent because tangent appears to be multiplying b but that would be incorrect. Trigonometry doesn't work exactly like Algebra, the way that we isolate b in this problem is by taking the inverse trig function. So since we have tangent of b we're going to say that if I took the inverse tangent of 14 twelves that's what b would be. To use b as many times as I can in one sentence, so the inverse tangent is looks like tangent to the negative 1. So in your calculator right above your tangent button is the inverse tangent. And you're probably going to have to put something like second to get there, so in my calculator I'm going to type inverse tangent of this fraction 14 to 12 and I get 49.4. So b equals 49.4 degrees. Let's look at 2 other quick examples, let's say I told you sine of x is equal to 0.5 what is x? To solve for x we're going take the inverse sign of both sides of this equation. So on the other side I'm going to write sine inverse of 0.5 so the inverse sine of the sine of x is x and that's the reason why we use that inverse property. So I'm going to type in inverse sine of 0.5 and I get 30 degrees. I still got one last one, if I said cosine of y is equal to 18 divided by 22 to solve for y we're going to take the inverse cosine. So we're going to take the inverse cosine of the cosine of y which will give us y and on the right side I'm going to have to take the inverse cosine of 18 20 seconds. So I'm going to say inverse cosine of 18 divided by 22 is 35.1. So whenever you need to solve for an angle when you're talking about a trig function you can get at that variable by taking the inverse function of whatever you're talking about sine, cosine or tangent.