In the triangle above we are going to solve for the missing angle measures and the missing side lengths. We can use the information from Target A-Pythagoras or Not because you are given two side lengths of a right triangle. With a little bit of review, you can find the length of \(\overline{JM}\). But what about the measure of \(\angle M\)? Or the measure of \(\angle J\)? Because the sum of each of the angles in the triangle is \(180^{\circ}\), we can only figure out that the \(m\angle J + m\angle M = 90^{\circ}\).
Inverse trigonometry is used to find missing angle measures of right triangles given two side lengths. You can only use inverse trigonometry with right triangles, as the similarity ratios do not exist without the right angle in each triangle.
Similar to right triangle trigonometry, there is such a thing as inverse sine (sin\(^{-1}\)), inverse cosine (cos\(^{-1}\)), and inverse tangent (tan\(^{-1}\)). When to use which will be based on which side lengths are given.
Inverse trigonometry is used to find missing angle measures of right triangles given two side lengths. You can only use inverse trigonometry with right triangles, as the similarity ratios do not exist without the right angle in each triangle.
Similar to right triangle trigonometry, there is such a thing as inverse sine (sin\(^{-1}\)), inverse cosine (cos\(^{-1}\)), and inverse tangent (tan\(^{-1}\)). When to use which will be based on which side lengths are given.
Please note: Just like the right triangle trig problems, your calculator needs to be in Degrees mode before completing any of the example problems.
Label the sides of the triangle with hypotenuse, opposite, and adjacent based on the angle you are trying to find. Let’s start by finding the \(m\angle R\).
Set up a trig equation based on what is given. In this case, since the adjacent side and hypotenuse lengths are given, you will use cosine and then inverse cosine: cos\((R) = \large\frac{10}{12}\) Now we can use the inverse cosine to solve for the angle measure. \(m\angle R = \text{cos}^{-1}\large\left(\frac{10}{12}\right) = 33.6^{\circ}\) Note: whenever you use the inverse notation, the ratio of sides goes in the parentheses rather than the angle measure. To find the \(m\angle W\), we can use the triangle sum theorem. Or, you can use inverse trigonometry again. Just to get more practice, we will set it up using inverse trig. Note that the we will label the opposite and adjacent side differently: \(m\angle W = \text{sin}^{-1}\large\left(\frac{10}{12}\right) = 56.4^{\circ}\) Here is a video of the solution: |
Solution:
Adam did not label the sides of his triangle correctly. In fact, he switched the hypotenuse and adjacent sides, which then made everything else afterwards incorrect. Here’s the new picture and solution:
Adam did not label the sides of his triangle correctly. In fact, he switched the hypotenuse and adjacent sides, which then made everything else afterwards incorrect. Here’s the new picture and solution:
Quick Check
1) If \(LM = 22\) units and \(EM = 13\) units, find the measure of each angle in the following triangle. Round each angle measure to the nearest tenth of a degree. 2) Which side lengths would you need in order to find the \(m\angle G\) using: a) inverse sine? b) inverse cosine? c) inverse tangent? |