WEBVTT
00:00:01.890 --> 00:00:11.080
In this video, we will learn how to find the missing angle in a right triangle using the appropriate inverse trigonometric function, given two side lengths.
00:00:11.520 --> 00:00:15.950
Letβs begin by recapping some of the vocabulary related to right triangles.
00:00:16.590 --> 00:00:21.400
Suppose we have a right triangle with one of the non-right angles labeled as π.
00:00:21.580 --> 00:00:25.830
The sides of the triangle have specific names in relation to this angle.
00:00:26.620 --> 00:00:34.160
The side of the triangle, which is directly opposite the right angle, which is always the longest side of a triangle, is called the hypotenuse.
00:00:35.000 --> 00:00:40.110
In relation to angle π, the side directly opposite this angle is called the opposite.
00:00:41.180 --> 00:00:47.350
And the final side of the triangle, which is between angle π and the right angle, is called the adjacent.
00:00:47.680 --> 00:00:55.260
Weβll often see the names for these three sides abbreviated to opp., adj., and hyp. or simply O, A, and H.
00:00:55.910 --> 00:01:03.810
The three trigonometric ratios sine, cosine, and tangent describe the ratios between different pairs of side lengths in a right triangle.
00:01:04.410 --> 00:01:12.210
For a fixed value of π, the ratio between each pair of side lengths is always the same, no matter how big the triangle is.
00:01:13.010 --> 00:01:18.880
We can use the acronym SOH CAH TOA to remember the definitions of the three trigonometric ratios.
00:01:19.120 --> 00:01:24.900
The first letter in each part refers to the trigonometric ratio either sine, cosine, or tangent.
00:01:25.530 --> 00:01:35.840
And then the next two letters refer to the pair of sides involved in that particular ratio, with the side in the numerator first, followed by the side in the denominator.
00:01:36.050 --> 00:01:39.970
So sin π is equal to the opposite divided by the hypotenuse.
00:01:40.600 --> 00:01:45.060
cos of π is the length of the adjacent divided by the length of the hypotenuse.
00:01:45.570 --> 00:01:49.780
And tangent or tan of π is the opposite divided by the adjacent.
00:01:50.080 --> 00:02:01.850
We should already be comfortable with using these three trigonometric ratios to calculate the length of a side in a right triangle, given the length of one of the other two sides and the measure of one of the non-right angles.
00:02:02.080 --> 00:02:08.270
In this video, weβll focus on finding the measure of an angle given the length of two of the triangleβs sides.
00:02:09.690 --> 00:02:13.080
To do this, weβll need to use the inverse trigonometric functions.
00:02:13.290 --> 00:02:18.640
These are essentially the functions that do the opposite of the sine, cosine, and tangent functions.
00:02:18.840 --> 00:02:27.280
We write them using the superscript negative one, and we read them as the inverse sine, inverse cosine, and inverse tangent functions.
00:02:27.460 --> 00:02:32.610
Theyβre also known as the arcsine, arc cosine, and arc tangent functions.
00:02:33.370 --> 00:02:38.820
It is important that we realize that this superscript of negative one does not mean the reciprocal.
00:02:39.010 --> 00:02:44.000
The inverse sin of π₯ is not the same as one over sin π₯.
00:02:44.760 --> 00:02:53.230
These inverse trigonometric functions are another way of describing the relationship between an angle and the values of its three trigonometric ratios.
00:02:53.480 --> 00:02:55.560
Their interpretation is as follows.
00:02:55.660 --> 00:03:06.490
If there is a value π₯ such that π₯ is equal to sin of π, then we can equivalently write this as π is equal to the inverse sin of π₯.
00:03:07.120 --> 00:03:14.660
In the same way, if there is a value π¦ such that π¦ is equal to the cos of π, then π is equal to the inverse cos of π¦.
00:03:14.880 --> 00:03:22.250
And if there is a value π§ such that π§ is equal to the tan of π, then π is equal to the inverse tan of π§.
00:03:23.010 --> 00:03:35.410
If we know the value of one of the three trigonometric ratios, so the value of either π₯, π¦, or π§, we can work backward to find the angle associated with this ratio by applying an inverse trigonometric function.
00:03:36.240 --> 00:03:45.030
To find these functions on our calculators, we usually have to press shift and then either the sin, cos, or tan button to get the inverse of each function.
00:03:45.540 --> 00:03:53.160
Weβll begin by looking at an example of how we can use these functions to find the measure of an angle given two side lengths in a right triangle.
00:03:54.750 --> 00:04:01.000
For the given figure, find the measure of angle π΅π΄πΆ in degrees to two decimal places.
00:04:02.090 --> 00:04:05.880
Letβs begin by identifying angle π΅π΄πΆ on the diagram.
00:04:06.160 --> 00:04:11.730
Itβs the angle formed when we travel from π΅ to π΄ to πΆ, so itβs this angle here.
00:04:12.490 --> 00:04:15.710
We can denote this using the Greek letter π if we wish.
00:04:15.950 --> 00:04:24.690
We see that the triangle weβre given is a right triangle, in which we know the lengths of two of the sides, and we wish to calculate the measure of one angle.
00:04:25.010 --> 00:04:28.670
We can therefore approach this problem using right triangle trigonometry.
00:04:29.570 --> 00:04:34.670
Weβll begin by labeling the three sides of the triangle in relation to this angle π.
00:04:35.370 --> 00:04:40.220
The longest side of the triangle, which is the side directly opposite the right angle, is the hypotenuse.
00:04:41.070 --> 00:04:45.770
The side opposite this angle π, thatβs side π΅πΆ, is the opposite.
00:04:46.540 --> 00:04:51.230
And the side between angle π and the right angle, side π΄π΅, is the adjacent.
00:04:52.120 --> 00:04:59.590
To help us decide which of the three trigonometric ratios we need in this question, we can recall the acronym SOH CAH TOA.
00:05:00.330 --> 00:05:04.990
The two side lengths weβre given for this triangle are the opposite and the adjacent.
00:05:04.990 --> 00:05:08.570
So this tells us it is the tangent ratio we need to use.
00:05:09.260 --> 00:05:18.920
For an angle π in a right triangle, the tan of angle π is defined to be equal to the length of the opposite side divided by the length of the adjacent.
00:05:19.530 --> 00:05:25.290
So for this triangle, we have that the tangent of angle π is equal to seven over five.
00:05:26.410 --> 00:05:30.970
To find the value of π, we need to apply the inverse tangent function.
00:05:31.820 --> 00:05:36.540
We have that π is equal to the inverse tan of seven over five.
00:05:36.930 --> 00:05:39.320
We can evaluate this on our calculators.
00:05:39.420 --> 00:05:45.290
We usually need to press shift and then the tan function to bring up the inverse tangent function.
00:05:46.160 --> 00:05:51.880
Weβve been asked to give our answer in degrees, so we must also make sure that our calculators are in degree mode.
00:05:52.320 --> 00:05:59.370
Evaluating the inverse tan of seven over five gives 54.4623 continuing.
00:06:00.110 --> 00:06:03.700
The question specifies that we should give our answer to two decimal places.
00:06:03.850 --> 00:06:09.700
And as the digit in the third decimal place is a two, we round down to 54.46.
00:06:10.500 --> 00:06:20.640
By applying the inverse tangent function then, we found that the measure of angle π΅π΄πΆ in degrees to two decimal places is 54.46 degrees.
00:06:22.160 --> 00:06:30.130
In our next example, weβll find both of the unknown angles in a right triangle using two different inverse trigonometric functions.
00:06:31.530 --> 00:06:38.590
For the given figure, find the measures of angle π΄π΅πΆ and angle π΄πΆπ΅ in degrees to two decimal places.
00:06:39.770 --> 00:06:44.870
Looking at the digram, we see that we have a right triangle in which we know the lengths of two sides.
00:06:45.030 --> 00:06:48.900
We can therefore approach this problem using right triangle trigonometry.
00:06:49.420 --> 00:06:53.260
Our first step in a problem like this is to label the sides of the triangle.
00:06:53.430 --> 00:06:58.790
But in order to do this, we need to know which angle weβre labeling the sides in relation to.
00:06:59.530 --> 00:07:03.840
Letβs calculate angle π΄π΅πΆ first, and we can label this angle as π.
00:07:04.500 --> 00:07:07.580
The side directly opposite this angle is the opposite.
00:07:08.210 --> 00:07:11.700
The side between this angle and the right angle is the adjacent.
00:07:12.280 --> 00:07:16.590
And the final side, which is always directly opposite the right angle is the hypotenuse.
00:07:17.380 --> 00:07:23.470
To decide which of the three trigonometric ratios we need to use, we can recall the acronym SOH CAH TOA.
00:07:24.250 --> 00:07:31.520
In relation to angle π΄π΅πΆ or angle π, the two sides whose length weβve been given are the adjacent and the hypotenuse.
00:07:31.520 --> 00:07:34.420
So it is the cosine ratio that we need to use.
00:07:35.070 --> 00:07:44.480
For an angle π in a right triangle, the cos of angle π is defined to be equal to the length of the adjacent side divided by the length of the hypotenuse.
00:07:45.240 --> 00:07:50.620
So substituting the lengths we know, we have the cos of π is equal to four-ninths.
00:07:51.500 --> 00:07:55.700
To find the value of π, we need to apply the inverse cosine function.
00:07:56.030 --> 00:08:01.110
This is the function that says if cos of π is four-ninths, what is the value of π?
00:08:01.930 --> 00:08:09.200
We can evaluate this on our calculators, usually by pressing shift and then the cos button to bring up the inverse cosine function.
00:08:09.480 --> 00:08:13.110
And it gives 63.6122 continuing.
00:08:13.840 --> 00:08:20.440
The question specifies that we should give our answer to two decimal places, so we round to 63.61 degrees.
00:08:21.310 --> 00:08:27.290
Next, we need to calculate the measure of angle π΄πΆπ΅, which we can label on our diagram as angle πΌ.
00:08:27.720 --> 00:08:33.660
Now, we could calculate this angle, using the fact that angles in a triangle sum to 180 degrees.
00:08:33.910 --> 00:08:40.790
But instead, weβll calculate this angle using trigonometry and then check our answer by summing the three angles.
00:08:41.420 --> 00:08:47.670
Now importantly, because weβre calculating a different angle, we need to relabel the sides in the triangle.
00:08:48.140 --> 00:08:51.300
The hypotenuse of a right triangle is always the same side.
00:08:51.300 --> 00:08:53.870
Itβs the side directly opposite the right angle.
00:08:54.130 --> 00:08:59.390
But the adjacent and the opposite sides are labeled in relation to the angle weβre calculating.
00:09:00.330 --> 00:09:04.010
The opposite is the side directly opposite this angle.
00:09:04.220 --> 00:09:09.460
So in relation to angle πΌ, itβs the side π΄π΅, which is the opposite.
00:09:10.230 --> 00:09:15.600
And then in relation to angle πΌ, itβs the side π΄πΆ, which is the adjacent.
00:09:16.450 --> 00:09:23.160
We now see that in relation to angle πΌ, it is the opposite and the hypotenuse whose lengths we know.
00:09:23.190 --> 00:09:26.060
And so this time, weβre going to need to use the sine ratio.
00:09:27.070 --> 00:09:35.350
For an angle πΌ in a right triangle, sin of πΌ is defined to be equal to the length of the opposite divided by the length of the hypotenuse.
00:09:36.000 --> 00:09:40.430
So this time, we have that sin of πΌ is equal to four-ninths.
00:09:41.210 --> 00:09:48.590
To find the value of πΌ, we need to apply the inverse sine function, giving πΌ is equal to the inverse sine of four-ninths.
00:09:49.370 --> 00:10:00.550
Evaluating this on a calculator, which must be in degree mode, gives 26.3877 continuing, and this rounds to 26.39 to two decimal places.
00:10:02.240 --> 00:10:10.090
We can check our answer by summing the measures of the three angles in the triangle and confirming that this is indeed equal to 180 degrees.
00:10:10.990 --> 00:10:27.630
So by applying two different trigonometric ratios and then their inverse trigonometric functions, we found the measure of angle π΄π΅πΆ is 63.61 degrees and the measure of angle π΄πΆπ΅ is 26.39 degrees, each to two decimal places.
00:10:29.230 --> 00:10:37.540
In the examples weβve seen so far, we found either one or two missing angles in a right triangle using the inverse trigonometric functions.
00:10:37.790 --> 00:10:45.270
Sometimes we may be required to go further than this and find all the missing angles and all the unknown side lengths in a right triangle.
00:10:45.700 --> 00:10:47.560
This is called solving a triangle.
00:10:47.710 --> 00:10:50.060
And weβll practice this in our next example.
00:10:51.250 --> 00:10:58.520
π΄π΅πΆ is a right triangle at π΅, where π΅πΆ equals 10 centimeters and π΄πΆ equals 18 centimeters.
00:10:58.820 --> 00:11:07.290
Find the length π΄π΅, giving the answer to the nearest centimeter, and the measure of angles π΄ and πΆ, giving the answer to the nearest degree.
00:11:08.030 --> 00:11:12.610
Letβs begin by sketching this triangle, which weβre told is a right triangle at π΅.
00:11:13.410 --> 00:11:18.850
The length of π΅πΆ is 10 centimeters and the length of π΄πΆ is 18 centimeters.
00:11:18.850 --> 00:11:24.430
We need to find the measures of both unknown angles and the length of the third side π΄π΅.
00:11:25.350 --> 00:11:30.510
Letβs begin by calculating the measure of angle π΄, which we can label as π on our diagram.
00:11:31.240 --> 00:11:38.220
As we have a right triangle in which we know two of the side lengths, we can calculate the measure of this angle using right triangle trigonometry.
00:11:38.610 --> 00:11:43.070
We begin by labeling the three sides of the triangle in relation to this angle.
00:11:43.420 --> 00:11:48.730
π΅πΆ is the opposite, π΄π΅ is the adjacent, and π΄πΆ is the hypotenuse.
00:11:49.540 --> 00:11:58.500
Recalling the acronym SOH CAH TOA, we can see that it is the sine ratio we need to use because the lengths weβve been given are the opposite and the hypotenuse.
00:11:59.200 --> 00:12:07.230
Through an angle π in a right triangle, the sin of angle π is defined to be equal to the length of the opposite divided by the length of the hypotenuse.
00:12:07.840 --> 00:12:13.530
So for this triangle, we have that sin of π is equal to 10 over 18.
00:12:14.470 --> 00:12:18.710
To find the value of π, we need to apply the inverse sine function.
00:12:18.920 --> 00:12:23.080
So we have that π is equal to the inverse sin of 10 over 18.
00:12:23.850 --> 00:12:30.900
Evaluating this on a calculator, which must be in degree mode, gives 33.7489 continuing.
00:12:31.030 --> 00:12:34.700
And then rounding to the nearest degree gives 34 degrees.
00:12:35.800 --> 00:12:37.820
So we found the measure of angle π΄.
00:12:37.820 --> 00:12:41.370
Now letβs consider how we could find the measure of angle πΆ.
00:12:42.200 --> 00:12:47.420
If we wish, we could relabel the sides of the triangle in relation to this angle.
00:12:47.610 --> 00:12:51.950
So π΄π΅ becomes the opposite, and π΅πΆ becomes the adjacent.
00:12:52.790 --> 00:12:56.640
We could then calculate the measure of angle πΆ using the cosine ratio.
00:12:57.600 --> 00:13:02.770
However, itβs more efficient to recall that angles in any triangle sum to 180 degrees.
00:13:02.960 --> 00:13:10.590
So to calculate the measure of the third angle in a triangle, we can subtract the measures of the other two angles from 180 degrees.
00:13:11.480 --> 00:13:17.090
This tells us that angle πΌ or angle πΆ to the nearest degree is 56 degrees.
00:13:17.850 --> 00:13:23.930
Finally, we need to calculate the length of side π΄π΅, which we can do using another trigonometric ratio.
00:13:24.820 --> 00:13:31.370
In relation to angle π or angle π΄ whose measure we know, the side π΄π΅ is the adjacent.
00:13:32.260 --> 00:13:42.140
Using the cosine ratio, we therefore have that the cos of 33.7489 continuing degrees is equal to π΄π΅ over 18.
00:13:42.830 --> 00:13:51.340
Multiplying both sides of this equation by 18 gives π΄π΅ is equal to 18 cos of 33.7489 degrees.
00:13:51.550 --> 00:13:54.960
And weβre using the unrounded value here for accuracy.
00:13:55.670 --> 00:14:03.760
Evaluating this on a calculator gives 14.9666 continuing, and rounding this to the nearest integer gives 15.
00:14:04.480 --> 00:14:09.140
We have then that the length of π΄π΅ to the nearest centimeter is 15 centimeters.
00:14:09.280 --> 00:14:15.750
And the measures of angles π΄ and πΆ, each to the nearest degree, are 34 degrees and 56 degrees.
00:14:16.740 --> 00:14:20.510
We can check our answer for the length of π΄π΅ using the Pythagorean theorem.
00:14:20.860 --> 00:14:27.220
In a right triangle, the sum of the squares of the two shorter sides is always equal to the square of the hypotenuse.
00:14:28.010 --> 00:14:37.380
If we take the unrounded value for π΄π΅ and square it and then add 10 squared for π΅πΆ squared, this gives 324.
00:14:38.100 --> 00:14:43.440
The square of the hypotenuse, thatβs 18 squared, is also equal to 324.
00:14:43.630 --> 00:14:49.000
And as these two values are the same, this confirms that our answer for π΄π΅ is correct.
00:14:49.680 --> 00:14:56.880
We could also have calculated the length of π΄π΅ by using the Pythagorean theorem and then checked our answer using trigonometry.
00:14:58.530 --> 00:15:01.210
Letβs now summarize the key points from this video.
00:15:02.310 --> 00:15:09.200
When working with right triangles, we use the terms opposite, adjacent, and hypotenuse to refer to the sides of the triangle.
00:15:09.640 --> 00:15:13.090
The hypotenuse is always opposite the right angle and is the longest side.
00:15:13.490 --> 00:15:19.200
The opposite and the adjacent are labeled in relation to a given angle, often denoted π.
00:15:20.060 --> 00:15:27.320
The opposite is the side directly opposite this angle, and the adjacent is the side between this angle and the right angle.
00:15:28.140 --> 00:15:34.150
The acronym SOH CAH TOA can help us remember the definitions of the three trigonometric ratios.
00:15:34.380 --> 00:15:45.080
sin of π is equal to the opposite over the hypotenuse, cos of π is equal to the adjacent over the hypotenuse, and tan of π is equal to the opposite over the adjacent.
00:15:46.000 --> 00:15:52.360
We can find the measure of an angle in a right triangle given two side lengths using the inverse trigonometric functions.
00:15:52.630 --> 00:15:59.080
If there is a value π₯ such that π₯ is equal to sin π, then π is equal to the inverse sin of π₯.
00:15:59.220 --> 00:16:04.340
If π¦ is equal to cos π, then π is equal to the inverse cos of π¦.
00:16:04.660 --> 00:16:10.040
And if π§ is equal to tan of π, then π is equal to the inverse tan of π§.
00:16:10.830 --> 00:16:21.520
We saw that we can use the three trigonometric functions and their inverses to solve triangles, which means to find the length of all unknown sides and the measures of all unknown angles.
00:16:21.720 --> 00:16:30.640
When doing this, we may also use the Pythagorean theorem or the angle sum in a triangle, either as an alternative method or to check our answers.