# Right Triangles and Trigonometry

# Inverse Trig Ratios

If we have a right triangle, we only need two more pieces of information and then we can calculate all the things.

- We need to know a side length. Doesn't matter if it's the hypotenuse or one of the legs. Any ol' side length will do.
- We need to know either another side length or one of the angle measurements. One that's not 90°, preferably.

### Sample Problem

For example, we have this triangle. If we want to know the length of the hypotenuse, finding a good angle and trig ratio to use should do the trick.

Plug in what we know.

Make sure the calculator is in degree mode, and then fire that baby up.

Hypotenuse ≈ 61.7

Armed with that knowledge, we can calculate the other leg of the triangle using the cosine.

However, using the leg we already have would be more accurate, so we'll stick with that.

Once again, we'll need a calculator.

Adjacent ≈ 48.6

Yippee. We now have all the side lengths and two out of the three angles. We can use the side lengths to figure out the remaining angle. Cosine hasn't been used yet and we don't want it to feel left out.

This time, we're looking for Y. The adjacent side we plug in has to be adjacent to the angle we're using.

cos *Y* = 0.616

Now what? Inverse cosine time.

That's right. *Inverse* cosine. If the cosine of *B* equals *y*, then the inverse cosine of *y* equals *B*. The same applies to sine and tangent, of course. These are called **inverse trig functions**, and they all look like this:

Normal Trig Function | Inverse Trig Function |

sin A = x | sin^{-1}(x) = A |

cos B = y | cos^{-1}(y) = B |

tan C = z | tan^{-1}(z) = C |

Sometimes, these inverse trig functions are written with "arc" in front of their names. For instance, "sin^{-1}(*x*)" turns into "arcsin(*x*)" and "cos^{-1}(*y*)" becomes "arcos(*y*)." There's no difference. They mean the exact same thing.

Applying these inverse trig functions to our problem, if cos(*Y*) = 0.616, then cos^{-1}(0.616) = *Y*. Surely our calculator can handle that. Again, make sure it's in degree mode.

cos^{-1}(0.616) = *Y**Y* = 52°

A quick way to double-check that answer would be to add up all the angles in the triangle. All the angles in any triangle should always add up to 180°. Always.

*X* + *Y* + *Z* = 180°

We know that right angle *Z* is 90° and we're given that angle *X* is 38°. We just calculated angle *Y* as 52°. We'll plug them in and cross our fingers.

38° + 52° + 90° = 180°

180° = 180°

Yessireebob. We got all our angles and all our side lengths using trigonometry.

### Sample Problem

Carl wants to relax in a hammock. Who doesn't, really? He's 6 feet away from one tree and 4 feet away from another. From where he's standing, he can see both trees are exactly 90° from each other. He wants us to find out the angles that the two trees make. (Don't make him do it! He's on vacation!)

Relative to Tree *A*, we know the opposite and adjacent sides. If we're looking for trig ratios to use (which, hint-hint, we are), tangent would be a good choice.

Since we're looking for the angle, we should use the *inverse* tangent.

Relative to Tree *A*, the opposite side is 3 feet and the adjacent side is 5 feet.

*A* ≈ 33.7°

Now for Tree *B*.

Notice that relative to Tree *B*, the sides are switched. The opposite side is now 5 feet and the adjacent side is now 3 feet.

*B* ≈ 56.3°

Carl appreciates our hard work, but he says that won't help much. He needs to know how long his hammock should be to hang it between the two trees. That means we need to figure out the distance between Tree *A* and Tree *B*.

We could use the Pythagorean theorem, but why cramp our style? Trigonometry has served us well, so let's keep at it. We can use either the sine or cosine of either Tree *A* or *B*. Any one of those four should give us the same value for the hypotenuse.

Hypotenuse ≈ 7.2 feet

Carl's hammock should be at least 7.2 feet long. That's a nice, big hammock.