Have you ever wondered how the roof of a building is designed? Or have you wondered how to measure the height of a tree without climbing it? These questions can all be answered by triangle trigonometry. By finding the angles and side lengths in triangles, people like architects can calculate the slope of a roof.
What is the Definition of Triangle Trigonometry?
First, let's look at what triangle trigonometry is.
Triangle trigonometry is a topic within mathematics that looks at the relationship between side lengths and angles in a triangle.
You might have heard of triangle identities and trigonometric identities, and wondered what the difference was.
A triangle identity is a rule that is true for all triangles. This includes rules like:
- the sum of the angles in a triangle is \(180^\circ\), also known as \(\pi /2\) radians; and
- the sum of any two sides is longer than the length of the remaining side.
On the other hand, trigonometric identities are ones that are true about ONLY right triangles. These include things like:
- the Law of Sines;
- the Law of Cosines;
- the Pythagorean Theorem;
- and many more!
Formulas in Triangle Trigonometry
Trigonometry formulas can help you find missing side lengths and missing angles in triangles. There are different formulas that are used depending on what information you have been given about the triangle.
First let's look at how you will usually label a right triangle.
The hypotenuse is the side that is opposite the right angle. It is always the side with the longest length on the triangle.
Choose an angle, and give it a name. In this case the picture below has it named \(\theta\). The opposite side is the side that is opposite the angle \(\theta\).
The adjacent side is the side that is next to, adjacent to, the angle \(\theta\).
Fig. 1. Labeling a triangle.
SOHCAHTOA
In triangle trigonometry there are different rules (also called trigonometric identities) that you can follow to allow you to use the correct trigonometric functions. When looking at a right triangle, you can label each side to help you identify which function is best to use, there is an acronym called SOHCAHTOA that can help you remember which function to use.
SOH - Sine equals Opposite over Hypotenuse
CAH - Cosine equals Adjacent over Hypotenuse
TOA - Tangent equals Opposite over Adjacent
SOHCAHTOA only applies to right triangles!
Formulas and SOHCAHTOA
Once you have labelled your triangle you are able to identify which function is best to use to help you find out information about the right triangle and also what information to substitute into the formula. Below you can see the six basic trigonometric functions and the formula for each of the functions:
\[ \begin{align} \sin \theta &= \frac {\mbox{opposite}}{\mbox{hypotenuse}} \\ \cos \theta &= \frac{\mbox{adjacent}}{\mbox{hypotenuse}}\\ \tan \theta &= \frac {\mbox{opposite}} {\mbox{adjacent}} \\ \csc \theta &= \frac{\mbox{hypotenuse}}{\mbox {opposite}}\\ \sec \theta &= \frac {\mbox{hypotenuse}}{\mbox{adjacent}} \\ \cot \theta &= \frac{\mbox{adjacent}}{\mbox{opposite}}\end{align} \]
For each function you can see that you need to input information from the labeled right triangle to find the value of the function.
The way in which the formula is used can be broken down into steps:
Step 1: Label the triangle.
Step 2: Chose the correct function.
Step 3: Input your variables from the triangle and solve for what you need.
Of course it is easier to see how to use these things in an example.
Find the value of the Sine function for the angle \( \theta\).
Fig. 2. Finding an angle in a triangle.
Answer:
Let's work through the steps.
Step 1: Label the triangle.You have been given the hypotenuse and the opposite, so label them on the diagram.
Fig. 3. Triangle labeled with hypotenuse and opposite sides.
Step 2: Chose the correct function.
When you have been given the opposite and hypotenuse you have everything you need to use the sine function.
\[ \sin \theta = \frac{\mbox{opposite}} {\mbox{hypotenuse}} \]
Step 3: Input your variables from the triangle, and solve for \(\sin \theta \).
In this case the opposite side is \( 6 \; \text{cm}\) and the hypotenuse is \( 8 \; \text{cm}\), so you have
\[ \sin \theta = \frac {6} {8} = \frac{3}{4}. \]
What about the area of a triangle?
Area of Triangles and Trigonometry
The area of a triangle is a way to talk about the space inside the three sides of the triangle. Take a look at Area of Triangle for more information and examples!
Applications of Right Triangle Trigonometry
Right triangle trigonometry can be applied to many different real-life scenarios. It can be used to help people understand distances. The heights of trees or the distance from the top of a cliff to the bottom can be measured using trigonometry if you know the angles. These angles are known as the angle of elevation or the angle of depression.
The angle of elevation is the angle from a horizontal line to an object above the line.
Let's take a look at an example.
Suppose you have a tree, and you want to know how tall it is. From the base of the tree you pace off \(100\) feet, and with your protractor you measure the angle to the top of the tree from your current position is \(60\) degrees. How tall is the tree?
Answer:
The top of the tree is above your position, so this problem involves an angle of elevation. In fact in this example the angle of elevation to the top of the tree is \(60\) degrees. Since you aren't given a diagram, you will need to draw your own and label it.
Fig. 4. Position of you relative to the tree.
Step 1: The distance to the tree is \(100\) feet, and the angle of elevation is \(60\) degrees. You are asked to find the height of the tree, which is the opposite side. For convenience give that side the name \(h\).
Step 2: Choose the correct function.
In the picture above you have the angle and the adjacent side, and are asked to find the opposite side. That involves the tangent function! Remember that
\[ \tan \theta = \frac{\mbox{opposite}} {\mbox{adjacent}}. \]
Step 3: Input your variables from the triangle, and solve for \(h\).
Putting in what you know,
\[ \tan 45 = \frac{h} {100}. \]
You can use what you know about Trigonometric Functions to get that
\[ \tan 45 = \frac{\sqrt{3}}{3} ,\]
so
\[ \frac{\sqrt{3}}{3} = \frac{h}{100}\]
and
\[ h = 100 \frac{\sqrt{3}}{3} \, \text{ft}. \]
That is an exact answer. You may be asked to find out approximately how tall the tree is, so you can plug that into a calculator to find that
\[ h \approx 57.7 \, \text{ft}.\]
How about the angle of depression?
The angle of depression is the angle from a horizontal line to an object below the line.
Let's take a look at an example.
You have gone climbing today! Your truck is parked at the bottom of the cliff, about \(200\) yards from the base of the cliff. It is a relatively sheer cliff, so pretty much a vertical climb. Once you reach the top of the cliff you estimate the angle of depression to your truck is about \(30\) degrees. How far up do you think you are?
Answer:
Step 1: It helps to draw a picture! Put in the information you know, like that your truck is \(200\) yards from the base of the cliff and the angle of depression is about \(30\) degrees. For convenience call the height of the cliff \(h\).
Fig. 5. Triangle showing you at the top of the cliff and the location of your truck.
Step 2: Choose the correct function.
You want to know how high the cliff is, in other words what is \(h\)? If you look at the placement of the angle, you have the opposite side of \(200\) yards, and you want to know the adjacent side. That means you will want to use the tangent function.
Step 3: Input your variables from the triangle, and solve for \(h\).
Using the fact that
\[ \tan \theta = \frac{\mbox{opposite}} {\mbox{adjacent}}, \]
you can put in the information you have to get
\[ \tan 30 = \frac{200} {h}. \]
You can use what you know about Trigonometric Functions to get that
\[ \tan 30 = \frac{\sqrt{3}}{3} ,\]
so
\[ \frac{\sqrt{3}}{3} = \frac{200}{h}\]
and
\[\begin{align} h &= \frac{200}{ \dfrac{\sqrt{3}}{3}} \\ &= \frac{200\cdot3}{\sqrt{3}} \\ &= \frac{600}{\sqrt{3}} \, \text{yd}. \end{align}\]
If you don't like the square root in the denominator, you can multiply the numerator and denominator by \(\sqrt{3}\) to get:
\[\begin{align} h &= \frac{600}{\sqrt{3}} \\ &= \frac{600\sqrt{3}}{(\sqrt{3})^2} \\ &= \frac{600\sqrt{3}}{3} \\ &= 200\sqrt{3} \, \text{yd}. \end{align}\]
That is an exact answer. You may be asked to find out approximately how far up you think you are, so you can plug that into a calculator to find that
\[ h \approx 346 \, \text{yd}.\]
Of course there can never be enough examples!
Triangle Trigonometry Examples
Sometimes you will be asked to find the values of all six of the trigonometric functions for a given angle.
Find the values of the six trigonometric functions about angle \(\theta\).
Fig. 6. Finding the six trigonometric values of an angle.
Answer:
As usual, first you will want to label the right triangle.
Fig. 7. Triangle with angle and sides labeled.
Then you can use SOHCHATOA to find the values of three of the trigonometric functions for the angle \(\theta\).
SOH:
\[ \begin{align} \sin \theta &= \frac {\mbox{opposite}}{\mbox{hypotenuse}} \\ &= \frac{8}{10} \\ &= \frac{4}{5}. \end{align}\]
CAH:
\[\begin{align} \cos \theta &= \frac{\mbox{adjacent}}{\mbox{hypotenuse}}\\ &= \frac{6}{10} \\ &= \frac{3}{5}.\end{align}\]
TOA:
\[ \begin{align} \tan \theta &= \frac {\mbox{opposite}} {\mbox{adjacent}} \\ &= \frac{8}{6} \\ &= \frac{4}{3}. \end{align} \]
Then for the other three,
Cosecant:
\[ \begin{align} \csc \theta &= \frac{\mbox{hypotenuse}}{\mbox {opposite}}\\ &= \frac{10}{8} \\ &= \frac{5}{4}. \end{align} \]
Secant:
\[ \begin{align} \sec \theta &= \frac {\mbox{hypotenuse}}{\mbox{adjacent}} \\ &= \frac{10}{6} \\ &= \frac{5}{3}. \end{align} \]
Cotangent:
\[ \begin{align} \cot \theta &= \frac{\mbox{adjacent}}{\mbox{opposite}} \\ &= \frac{6}{8} \\ &= \frac{3}{4}. \end{align} \]
Sometimes you will be asked to find a missing side.
Find \(x\).
Fig. 8. Triangle with given angle and opposite side.
Answer:Of course the joke is to just draw an arrow to the letter \(x\) in the picture! However what this question is actually asking you to do is find the measurement of \(x\) in centimeters.
Step 1: Label the triangle. It is already done!
Step 2: Chose the correct function.
The information that you have is an angle and the opposite side, and what you want to find is the hypotenuse. That means you will want to use the sine function.
Step 3: Input your variables from the triangle, and solve for \(x\).
Using the SOH part of SOHCAHTOA, you know that
\[\sin \theta = \frac{\mbox{opposite}}{\mbox {hypotenuse}} .\]
Plugging in what you know gives you
\[\sin 55 = \frac{16}{x} ,\]
so
\[ x = \frac{ 16}{\sin 55} \, \text{cm}. \]
If you are asked to find the approximate value of \(x\) to two decimal places, the answer would be
\[ x = 19.53 \, \text{cm}.\]
Sometimes you will be asked to "solve" a triangle. What this is really asking you do do is find all of the angles and all of the sides.
Solve the triangle \(\Delta ABC\).
Fig. 9. Solving the triangle.
Answer:
This triangle is already helpfully labeled! You know that the angle at corner \(B\) is \(90^\circ\), you have two of the sides, and one of the other angles. That just leaves you to find the length of side \(x\) and the measurement of angle \(y\).
Let's look at finding \(x\) first. It is the hypotenuse, you have the angle at corner \(A\) being \(50^\circ\), and the side adjacent has length \(5\, \text{cm}\). So you will want to use the formula
\[\cos \theta = \frac{\mbox{adjacent}}{\mbox{hypotenuse}}.\]
Plugging in what you know,
\[ \cos 50= \frac{5}{x}, \]
so
\[ x = \frac{5}{\cos 50} \, \text{cm}.\]
Next let's find \(y\). To do this you can use a simple subtraction since you know all the angles in a triangle sum up to \(180°\). That gives you
\[ 180 = y + 55 + 90\]
so solving for \(y\),
\[ y = 35^\circ .\]
Triangle trigonometry - Key takeaways
- Triangle trigonometry looks at the relationship between angles and sides on a triangle.
- There are 6 different functions that are used in trigonometry; sine, cosine, tangent, cosecant, secant and cotangent.
- The acronym SOHCAHTOA can be used to remind you which function is appropriate to use when solving a right triangle.
SOH - Sine equals Opposite over Hypotenuse
CAH - Cosine equals Adjacent over Hypotenuse
TOA - Tangent equals Opposite over Adjacent
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