Video Transcript
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.
Let’s begin by recapping some of
the vocabulary related to right triangles. Suppose we have a right triangle
with one of the non-right angles labeled as 𝜃. The sides of the triangle have
specific names in relation to this angle. 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. In relation to angle 𝜃, the side
directly opposite this angle is called the opposite. And the final side of the triangle,
which is between angle 𝜃 and the right angle, is called the adjacent. We’ll often see the names for these
three sides abbreviated to opp., adj., and hyp. or simply O, A, and H.
The three trigonometric ratios
sine, cosine, and tangent describe the ratios between different pairs of side
lengths in a right triangle. For a fixed value of 𝜃, the ratio
between each pair of side lengths is always the same, no matter how big the triangle
is. We can use the acronym SOH CAH TOA
to remember the definitions of the three trigonometric ratios. The first letter in each part
refers to the trigonometric ratio either sine, cosine, or tangent. 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. So sin 𝜃 is equal to the opposite
divided by the hypotenuse. cos of 𝜃 is the length of the
adjacent divided by the length of the hypotenuse. And tangent or tan of 𝜃 is the
opposite divided by the adjacent.
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. In this video, we’ll focus on
finding the measure of an angle given the length of two of the triangle’s sides. To do this, we’ll need to use the
inverse trigonometric functions. These are essentially the functions
that do the opposite of the sine, cosine, and tangent functions. We write them using the superscript
negative one, and we read them as the inverse sine, inverse cosine, and inverse
tangent functions. They’re also known as the arc sine,
arc cosine, and arc tangent functions.
It is important that we realize
that this superscript of negative one does not mean the reciprocal. The inverse sin of 𝑥 is not the
same as one over sin 𝑥. These inverse trigonometric
functions are another way of describing the relationship between an angle and the
values of its three trigonometric ratios. Their interpretation is as
follows. If there is a value 𝑥 such that 𝑥
is equal to sin of 𝜃, then we can equivalently write this as 𝜃 is equal to the
inverse sin of 𝑥. In the same way, if there is a
value 𝑦 such that 𝑦 is equal to the cos of 𝜃, then 𝜃 is equal to the inverse cos
of 𝑦. And if there is a value 𝑧 such
that 𝑧 is equal to the tan of 𝜃, then 𝜃 is equal to the inverse tan of 𝑧.
If we know the value of one of the
three trigonometric ratios, so the value of either 𝑥, 𝑦, or 𝑧, we can work
backward to find the angle associated with this ratio by applying an inverse
trigonometric function. 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. We’ll 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.
For the given figure, find the
measure of angle 𝐵𝐴𝐶 in degrees to two decimal places.
Let’s begin by identifying angle
𝐵𝐴𝐶 on the diagram. It’s the angle formed when we
travel from 𝐵 to 𝐴 to 𝐶, so it’s this angle here. We can denote this using the Greek
letter 𝜃 if we wish. We see that the triangle we’re
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. We can therefore approach this
problem using right triangle trigonometry.
We’ll begin by labeling the three
sides of the triangle in relation to this angle 𝜃. The longest side of the triangle,
which is the side directly opposite the right angle, is the hypotenuse. The side opposite this angle 𝜃,
that’s side 𝐵𝐶, is the opposite. And the side between angle 𝜃 and
the right angle, side 𝐴𝐵, is the adjacent. To help us decide which of the
three trigonometric ratios we need in this question, we can recall the acronym SOH
CAH TOA. The two side lengths we’re given
for this triangle are the opposite and the adjacent. So this tells us it is the tangent
ratio we need to use.
For an angle 𝜃 in a right
triangle, the tan of angle 𝜃 is defined to be equal to the length of the opposite
side divided by the length of the adjacent. So for this triangle, we have that
the tangent of angle 𝜃 is equal to seven over five. To find the value of 𝜃, we need to
apply the inverse tangent function. We have that 𝜃 is equal to the
inverse tan of seven over five. We can evaluate this on our
calculators. We usually need to press shift and
then the tan function to bring up the inverse tangent function.
We’ve been asked to give our answer
in degrees, so we must also make sure that our calculators are in degree mode. Evaluating the inverse tan of seven
over five gives 54.4623 continuing. The question specifies that we
should give our answer to two decimal places. And as the digit in the third
decimal place is a two, we round down to 54.46. By applying the inverse tangent
function then, we found that the measure of angle 𝐵𝐴𝐶 in degrees to two decimal
places is 54.46 degrees.
In our next example, we’ll find
both of the unknown angles in a right triangle using two different inverse
trigonometric functions.
For the given figure, find the
measures of angle 𝐴𝐵𝐶 and angle 𝐴𝐶𝐵 in degrees to two decimal places.
Looking at the digram, we see that
we have a right triangle in which we know the lengths of two sides. We can therefore approach this
problem using right triangle trigonometry. Our first step in a problem like
this is to label the sides of the triangle. But in order to do this, we need to
know which angle we’re labeling the sides in relation to. Let’s calculate angle 𝐴𝐵𝐶 first,
and we can label this angle as 𝜃. The side directly opposite this
angle is the opposite. The side between this angle and the
right angle is the adjacent. And the final side, which is always
directly opposite the right angle is the hypotenuse.
To decide which of the three
trigonometric ratios we need to use, we can recall the acronym SOH CAH TOA. In relation to angle 𝐴𝐵𝐶 or
angle 𝜃, the two sides whose length we’ve been given are the adjacent and the
hypotenuse. So it is the cosine ratio that we
need to use. For an angle 𝜃 in a right
triangle, the cos of angle 𝜃 is defined to be equal to the length of the adjacent
side divided by the length of the hypotenuse. So substituting the lengths we
know, we have the cos of 𝜃 is equal to four-ninths. To find the value of 𝜃, we need to
apply the inverse cosine function. This is the function that says if
cos of 𝜃 is four-ninths, what is the value of 𝜃?
We can evaluate this on our
calculators, usually by pressing shift and then the cos button to bring up the
inverse cosine function. And it gives 63.6122
continuing. The question specifies that we
should give our answer to two decimal places, so we round to 63.61 degrees. Next, we need to calculate the
measure of angle 𝐴𝐶𝐵, which we can label on our diagram as angle 𝛼. Now, we could calculate this angle,
using the fact that angles in a triangle sum to 180 degrees. But instead, we’ll calculate this
angle using trigonometry and then check our answer by summing the three angles.
Now importantly, because we’re
calculating a different angle, we need to relabel the sides in the triangle. The hypotenuse of a right triangle
is always the same side. It’s the side directly opposite the
right angle. But the adjacent and the opposite
sides are labeled in relation to the angle we’re calculating. The opposite is the side directly
opposite this angle. So in relation to angle 𝛼, it’s
the side 𝐴𝐵, which is the opposite. And then in relation to angle 𝛼,
it’s the side 𝐴𝐶, which is the adjacent.
We now see that in relation to
angle 𝛼, it is the opposite and the hypotenuse whose lengths we know. And so this time, we’re going to
need to use the sine ratio. For an angle 𝛼 in a right
triangle, sin of 𝛼 is defined to be equal to the length of the opposite divided by
the length of the hypotenuse. So this time, we have that sin of
𝛼 is equal to four-ninths. To find the value of 𝛼, we need to
apply the inverse sine function, giving 𝛼 is equal to the inverse sine of
four-ninths. 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.
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. So by applying two different
trigonometric ratios and then their inverse trigonometric functions, we found the
measure of angle 𝐴𝐵𝐶 is 63.61 degrees and the measure of angle 𝐴𝐶𝐵 is 26.39
degrees, each to two decimal places.
In the examples we’ve seen so far,
we found either one or two missing angles in a right triangle using the inverse
trigonometric functions. 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. This is called solving a
triangle. And we’ll practice this in our next
example.
𝐴𝐵𝐶 is a right triangle at 𝐵,
where 𝐵𝐶 equals 10 centimeters and 𝐴𝐶 equals 18 centimeters. Find the length 𝐴𝐵, giving the
answer to the nearest centimeter, and the measure of angles 𝐴 and 𝐶, giving the
answer to the nearest degree.
Let’s begin by sketching this
triangle, which we’re told is a right triangle at 𝐵. The length of 𝐵𝐶 is 10
centimeters and the length of 𝐴𝐶 is 18 centimeters. We need to find the measures of
both unknown angles and the length of the third side 𝐴𝐵. Let’s begin by calculating the
measure of angle 𝐴, which we can label as 𝜃 on our diagram. 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. We begin by labeling the three
sides of the triangle in relation to this angle. 𝐵𝐶 is the opposite, 𝐴𝐵 is the
adjacent, and 𝐴𝐶 is the hypotenuse.
Recalling the acronym SOH CAH TOA,
we can see that it is the sine ratio we need to use because the lengths we’ve been
given are the opposite and the hypotenuse. Through an angle 𝜃 in a right
triangle, the sin of angle 𝜃 is defined to be equal to the length of the opposite
divided by the length of the hypotenuse. So for this triangle, we have that
sin of 𝜃 is equal to 10 over 18. To find the value of 𝜃, we need to
apply the inverse sine function. So we have that 𝜃 is equal to the
inverse sin of 10 over 18. Evaluating this on a calculator,
which must be in degree mode, gives 33.7489 continuing. And then rounding to the nearest
degree gives 34 degrees.
So we found the measure of angle
𝐴. Now let’s consider how we could
find the measure of angle 𝐶. If we wish, we could relabel the
sides of the triangle in relation to this angle. So 𝐴𝐵 becomes the opposite, and
𝐵𝐶 becomes the adjacent. We could then calculate the measure
of angle 𝐶 using the cosine ratio. However, it’s more efficient to
recall that angles in any triangle sum to 180 degrees. 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. This tells us that angle 𝛼 or
angle 𝐶 to the nearest degree is 56 degrees.
Finally, we need to calculate the
length of side 𝐴𝐵, which we can do using another trigonometric ratio. In relation to angle 𝜃 or angle 𝐴
whose measure we know, the side 𝐴𝐵 is the adjacent. Using the cosine ratio, we
therefore have that the cos of 33.7489 continuing degrees is equal to 𝐴𝐵 over
18. Multiplying both sides of this
equation by 18 gives 𝐴𝐵 is equal to 18 cos of 33.7489 degrees. And we’re using the unrounded value
here for accuracy. Evaluating this on a calculator
gives 14.9666 continuing, and rounding this to the nearest integer gives 15. We have then that the length of
𝐴𝐵 to the nearest centimeter is 15 centimeters. And the measures of angles 𝐴 and
𝐶, each to the nearest degree, are 34 degrees and 56 degrees.
We can check our answer for the
length of 𝐴𝐵 using the Pythagorean theorem. In a right triangle, the sum of the
squares of the two shorter sides is always equal to the square of the
hypotenuse. If we take the unrounded value for
𝐴𝐵 and square it and then add 10 squared for 𝐵𝐶 squared, this gives 324. The square of the hypotenuse,
that’s 18 squared, is also equal to 324. And as these two values are the
same, this confirms that our answer for 𝐴𝐵 is correct. We could also have calculated the
length of 𝐴𝐵 by using the Pythagorean theorem and then checked our answer using
trigonometry.
Let’s now summarize the key points
from this video. When working with right triangles,
we use the terms opposite, adjacent, and hypotenuse to refer to the sides of the
triangle. The hypotenuse is always opposite
the right angle and is the longest side. The opposite and the adjacent are
labeled in relation to a given angle, often denoted 𝜃. The opposite is the side directly
opposite this angle, and the adjacent is the side between this angle and the right
angle.
The acronym SOH CAH TOA can help us
remember the definitions of the three trigonometric ratios. sin of 𝜃 is equal to the opposite
over the hypotenuse, cos of 𝜃 is equal to the adjacent over the hypotenuse, and tan
of 𝜃 is equal to the opposite over the adjacent. We can find the measure of an angle
in a right triangle given two side lengths using the inverse trigonometric
functions. If there is a value 𝑥 such that 𝑥
is equal to sin 𝜃, then 𝜃 is equal to the inverse sin of 𝑥. If 𝑦 is equal to cos 𝜃, then 𝜃
is equal to the inverse cos of 𝑦. And if 𝑧 is equal to tan of 𝜃,
then 𝜃 is equal to the inverse tan of 𝑧.
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. 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.
