 Section 8.4 also deals with inverse trigonometry. We use inverse trigonometry when we want to find the angle measure of a right triangle given two side lengths. So first, we need a right triangle with two given side lengths at a missing angle. We'll set up a trig ratio equation, and then we'll solve for the unknown angle using inverse trigonometry. So, inverses. You probably remember inverses from chapter 2 when we were dealing with logic statements. Inverse trigonometry is not the same, well it's the same kind of idea but we're not talking about if not p then not q. Instead, inverse is a function that reverses another function. So for example, the inverse of addition is subtraction. The inverse of multiplication is division. The inverse of squaring a number is to take the square root of the number. And so therefore the inverse of sine is literally called inverse sine. It's oftentimes shown as sine raised to the negative one power. It's not actually sine raised to the negative one power, it's the inverse sine. So your calculator has inverse trig functions. You'll just need to tap second and then sine, second cosine, second tangent, that kind of stuff. So some examples here. Take a minute and jot these down. We'll work through them one by one. The first equation, we have tangent of angle A is 16 11ths. So that means A is the inverse tangent of 16 11ths. And so then pulling out your calculator, type second tangent 16 11ths, which gives us about 55.49. With the TI 30, you'll want to type in 16 11ths first and then take the inverse tangent of that. The next example, the sine of B. So therefore B is the inverse sine of 21 43rds, which is 29.23 degrees. And then the last one, cosine, we'll use the inverse cosine. However, when you type that in, you get a problem. Inverse cosine of 11 fifths is not defined. Why would that be? Well, think about cosine. Cosine is adjacent over hypotenuse. And so we're saying that 5 is the hypotenuse of a right triangle. That's impossible for 11 to be longer than 5. The hypotenuse is always the longest side in a right triangle. So therefore that's kind of a bogus question. All right, let's try an example with a triangle. Here we have 6 and 8 are given. We want to find angle A. And so therefore, if we know the opposite and we know the adjacent, we'll use the tangent function. So the tangent of angle A is 6 eighths. Whenever we want to solve for an angle, remember we're using inverse trig. And so A is the inverse tangent of 6 eighths. Now 6 eighths, if you chose, you could simplify that to 3 fourths or 0.75. Regardless, when using your calculator, inverse tangent of let's say 0.75 is 36.87. Likewise, you could just use 6 over 8. So we get 36.87 degrees. Here we have a few more examples. In that first box, we've got angle B is unknown. The opposite 5, the hypotenuse is 13. Opposite and hypotenuse means we'll use the sine function. So the sine of the angle is 5 thirteenths, which means the angle is the inverse sine of 5 thirteenths. And so we get an angle measure of about 22.62 degrees. The next example, we're given the adjacent and hypotenuse of angle C. And so that means we'll use the inverse cosine function. Inverse cosine of adjacent divided by hypotenuse gives us about 61.93 degrees. This third example, we are given the opposite and hypotenuse for angle D. And so therefore D is the inverse sine of 3 over 3 root 2, which is, hey, 45 degrees. That's a special right triangle. Last example, we're given angle G and so forth with its opposite and adjacent. We should use the tangent function. So the tangent of the angle 40 over 9. And therefore G is the inverse tangent of 40 over 9. We get 77.32 degrees.