## Right Triangles and the Pythagorean Theorem

The Pythagorean Theorem, ${\displaystyle a^{2}+b^{2}=c^{2},}$ can be used to find the length of any side of a right triangle.

### Learning Objectives

Use the Pythagorean Theorem to find the length of a side of a right triangle

### Key Takeaways

#### Key Points

• The Pythagorean Theorem, ${\displaystyle a^{2}+b^{2}=c^{2},}$ is used to find the length of any side of a right triangle.
• In a right triangle, one of the angles has a value of 90 degrees.
• The longest side of a right triangle is called the hypotenuse, and it is the side that is opposite the 90 degree angle.
• If the length of the hypotenuse is labeled $c$, and the lengths of the other sides are labeled $a$ and $b$, the Pythagorean Theorem states that ${\displaystyle a^{2}+b^{2}=c^{2}}$.

#### Key Terms

• legs: The sides adjacent to the right angle in a right triangle.
• right triangle: A $3$-sided shape where one angle has a value of $90$ degrees
• hypotenuse: The side opposite the right angle of a triangle, and the longest side of a right triangle.
• Pythagorean theorem: The sum of the areas of the two squares on the legs ($a$ and $b$) is equal to the area of the square on the hypotenuse ($c$). The formula is $a^2+b^2=c^2$.

### Right Triangle

A right angle has a value of 90 degrees ($90^\circ$). A right triangle is a triangle in which one angle is a right angle. The relation between the sides and angles of a right triangle is the basis for trigonometry.

The side opposite the right angle is called the hypotenuse (side $c$ in the figure). The sides adjacent to the right angle are called legs (sides $a$ and $b$). Side $a$ may be identified as the side adjacent to angle $B$ and opposed to (or opposite) angle $A$. Side $b$ is the side adjacent to angle $A$ and opposed to angle $B$.

Right triangle: The Pythagorean Theorem can be used to find the value of a missing side length in a right triangle.

If the lengths of all three sides of a right triangle are whole numbers, the triangle is said to be a Pythagorean triangle and its side lengths are collectively known as a Pythagorean triple.

### The Pythagorean Theorem

The Pythagorean Theorem, also known as Pythagoras’ Theorem, is a fundamental relation in Euclidean geometry. It defines the relationship among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem can be written as an equation relating the lengths of the sides $a$, $b$ and $c$, often called the “Pythagorean equation”:[1]

${\displaystyle a^{2}+b^{2}=c^{2}}$

In this equation, $c$ represents the length of the hypotenuse and $a$ and $b$ the lengths of the triangle’s other two sides.

Although it is often said that the knowledge of the theorem predates him,[2] the theorem is named after the ancient Greek mathematician Pythagoras (c. 570 – c. 495 BC). He is credited with its first recorded proof.

The Pythagorean Theorem: The sum of the areas of the two squares on the legs ($a$ and $b$) is equal to the area of the square on the hypotenuse ($c$).  The formula is $a^2+b^2=c^2$.

### Finding a Missing Side Length

Example 1:  A right triangle has a side length of $10$ feet, and a hypotenuse length of $20$ feet.  Find the other side length.  (round to the nearest tenth of a foot)

Substitute $a=10$ and $c=20$ into the Pythagorean Theorem and solve for $b$.

\displaystyle{ \begin{align} a^{2}+b^{2} &=c^{2} \\ (10)^2+b^2 &=(20)^2 \\ 100+b^2 &=400 \\ b^2 &=300 \\ \sqrt{b^2} &=\sqrt{300} \\ b &=17.3 ~\mathrm{feet} \end{align} }

Example 2:  A right triangle has side lengths $3$ cm and $4$ cm.  Find the length of the hypotenuse.

Substitute $a=3$ and $b=4$ into the Pythagorean Theorem and solve for $c$.

\displaystyle{ \begin{align} a^{2}+b^{2} &=c^{2} \\ 3^2+4^2 &=c^2 \\ 9+16 &=c^2 \\ 25 &=c^2\\ c^2 &=25 \\ \sqrt{c^2} &=\sqrt{25} \\ c &=5~\mathrm{cm} \end{align} }

## How Trigonometric Functions Work

Trigonometric functions can be used to solve for missing side lengths in right triangles.

### Learning Objectives

Recognize how trigonometric functions are used for solving problems about right triangles, and identify their inputs and outputs

### Key Takeaways

#### Key Points

• A right triangle has one angle with a value of 90 degrees ($90^{\circ}$)The three trigonometric functions most often used to solve for a missing side of a right triangle are: $\displaystyle{\sin{t}=\frac {opposite}{hypotenuse}}$, $\displaystyle{\cos{t} = \frac {adjacent}{hypotenuse}}$, and $\displaystyle{\tan{t} = \frac {opposite}{adjacent}}$

### Trigonometric Functions

We can define the trigonometric functions in terms an angle $t$ and the lengths of the sides of the triangle. The adjacent side is the side closest to the angle. (Adjacent means “next to.”) The opposite side is the side across from the angle. The hypotenuse  is the side of the triangle opposite the right angle, and it is the longest.

Right triangle: The sides of a right triangle in relation to angle $t$.

When solving for a missing side of a right triangle, but the only given information is an acute angle measurement and a side length, use the trigonometric functions listed below:

• Sine           $\displaystyle{\sin{t} = \frac {opposite}{hypotenuse}}$
• Cosine       $\displaystyle{\cos{t} = \frac {adjacent}{hypotenuse}}$
• Tangent    $\displaystyle{\tan{t} = \frac {opposite}{adjacent}}$

The trigonometric functions are equal to ratios that relate certain side lengths of a  right triangle.  When solving for a missing side, the first step is to identify what sides and what angle are given, and then select the appropriate function to use to solve the problem.

### Evaluating a Trigonometric Function of a Right Triangle

Sometimes you know the length of one side of a triangle and an angle, and need to find other measurements.  Use one of the trigonometric functions ($\sin{}$, $\cos{}$, $\tan{}$), identify the sides and angle given, set up the equation and use the calculator and algebra to find the missing side length.

Example 1:
Given a right triangle with acute angle of $34^{\circ}$ and a hypotenuse length of $25$ feet, find the length of the side opposite the acute angle (round to the nearest tenth):

Right triangle: Given a right triangle with acute angle of $34$ degrees and a hypotenuse length of $25$ feet, find the opposite side length.

Looking at the figure, solve for the side opposite the acute angle of $34$ degrees.  The ratio of the sides would be the opposite side and the hypotenuse.  The ratio that relates those two sides is the sine function.

\displaystyle{ \begin{align} \sin{t} &=\frac {opposite}{hypotenuse} \\ \sin{\left(34^{\circ}\right)} &=\frac{x}{25} \\ 25\cdot \sin{ \left(34^{\circ}\right)} &=x\\ x &= 25\cdot \sin{ \left(34^{\circ}\right)}\\ x &= 25 \cdot \left(0.559\dots\right)\\ x &=14.0 \end{align} }

The side opposite the acute angle is $14.0$ feet.

Example 2:
Given a right triangle with an acute angle of $83^{\circ}$ and a hypotenuse length of $300$ feet, find the hypotenuse length (round to the nearest tenth):

Right Triangle: Given a right triangle with an acute angle of $83$ degrees and a hypotenuse length of $300$ feet, find the hypotenuse length.

Looking at the figure, solve for the hypotenuse to the acute angle of $83$ degrees. The ratio of the sides would be the adjacent side and the hypotenuse.  The ratio that relates these two sides is the cosine function.

\displaystyle{ \begin{align} \cos{t} &= \frac {adjacent}{hypotenuse} \\ \cos{ \left( 83 ^{\circ}\right)} &= \frac {300}{x} \\ x \cdot \cos{\left(83^{\circ}\right)} &=300 \\ x &=\frac{300}{\cos{\left(83^{\circ}\right)}} \\ x &= \frac{300}{\left(0.1218\dots\right)} \\ x &=2461.7~\mathrm{feet} \end{align} }

## Sine, Cosine, and Tangent

The mnemonic
SohCahToa can be used to solve for the length of a side of a right triangle.

### Learning Objectives

Use the acronym SohCahToa to define Sine, Cosine, and Tangent in terms of right triangles

### Key Takeaways

#### Key Points

• A common mnemonic for remembering the relationships between the Sine, Cosine, and Tangent functions is SohCahToa.
• SohCahToa is formed from the first letters of “Sine is opposite over hypotenuse (Soh), Cosine is adjacent over hypotenuse (Cah), Tangent is opposite over adjacent (Toa).”

### Definitions of Trigonometric Functions

Given a right triangle with an acute angle of $t$, the first three trigonometric functions are:

• Sine             $\displaystyle{ \sin{t} = \frac {opposite}{hypotenuse} }$
• Cosine        $\displaystyle{ \cos{t} = \frac {adjacent}{hypotenuse} }$
• Tangent      $\displaystyle{ \tan{t} = \frac {opposite}{adjacent} }$

A common mnemonic for remembering these relationships is SohCahToa, formed from the first letters of “Sine is opposite over hypotenuse (Soh), Cosine is adjacent over hypotenuse (Cah), Tangent is opposite over adjacent (Toa).”

Right triangle: The sides of a right triangle in relation to angle $t$. The hypotenuse is the long side, the opposite side is across from angle $t$, and the adjacent side is next to angle $t$.

### Evaluating a Trigonometric Function of a Right Triangle

Example 1:
Given a right triangle with an acute angle of $62^{\circ}$ and an adjacent side of $45$ feet, solve for the opposite side length. (round to the nearest tenth)

Right triangle: Given a right triangle with an acute angle of $62$ degrees and an adjacent side of $45$ feet, solve for the opposite side length.

First, determine which trigonometric function to use when given an adjacent side, and you need to solve for the opposite side.  Always determine which side is given and which side is unknown from the acute angle ($62$ degrees).  Remembering the mnemonic, “SohCahToa”, the sides given are opposite and adjacent or “o” and “a”, which would use “T”, meaning the tangent trigonometric function.

\displaystyle{ \begin{align} \tan{t} &= \frac {opposite}{adjacent} \\ \tan{\left(62^{\circ}\right)} &=\frac{x}{45} \\ 45\cdot \tan{\left(62^{\circ}\right)} &=x \\ x &= 45\cdot \tan{\left(62^{\circ}\right)}\\ x &= 45\cdot \left( 1.8807\dots \right) \\ x &=84.6 \end{align} }

Example 2:  A ladder with a length of $30~\mathrm{feet}$ is leaning against a building.  The angle the ladder makes with the ground is $32^{\circ}$.  How high up the building does the ladder reach? (round to the nearest tenth of a foot)

Right triangle: After sketching a picture of the problem, we have the triangle shown. The angle given is $32^\circ$, the hypotenuse is 30 feet, and the missing side length is the opposite leg, $x$ feet.

Determine which trigonometric function to use when given the hypotenuse, and you need to solve for the opposite side.  Remembering the mnemonic, “SohCahToa”, the sides given are the hypotenuse and opposite or “h” and “o”, which would use “S” or the sine trigonometric function.

\displaystyle{ \begin{align} \sin{t} &= \frac {opposite}{hypotenuse} \\ \sin{ \left( 32^{\circ} \right) } & =\frac{x}{30} \\ 30\cdot \sin{ \left(32^{\circ}\right)} &=x \\ x &= 30\cdot \sin{ \left(32^{\circ}\right)}\\ x &= 30\cdot \left( 0.5299\dots \right) \\ x &= 15.9 ~\mathrm{feet} \end{align} }

## Finding Angles From Ratios: Inverse Trigonometric Functions

The inverse trigonometric functions can be used to find the acute angle measurement of a right triangle.

### Learning Objectives

Use inverse trigonometric functions in solving problems involving right triangles

### Key Takeaways

#### Key Points

• A missing acute angle value of a right triangle can be found when given two side lengths.
• To find a missing angle value, use the trigonometric functions sine, cosine, or tangent, and the inverse key on a calculator to apply the inverse function ($\arcsin{}$, $\arccos{}$, $\arctan{}$), $\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$.

Using the trigonometric functions to solve for a missing side when given an acute angle is as simple as identifying the sides in relation to the acute angle, choosing the correct function, setting up the equation and solving.  Finding the missing acute angle when given two sides of a right triangle is just as simple.

### Inverse Trigonometric Functions

In order to solve for the missing acute angle, use the same three trigonometric functions, but, use the inverse key ($^{-1}$on the calculator) to solve for the angle ($A$) when given two sides.

$\displaystyle{ A^{\circ} = \sin^{-1}{ \left( \frac {\text{opposite}}{\text{hypotenuse}} \right) } }$

$\displaystyle{ A^{\circ} = \cos^{-1}{ \left( \frac {\text{adjacent}}{\text{hypotenuse}} \right) } }$

$\displaystyle{ A^{\circ} = \tan^{-1}{\left(\frac {\text{opposite}}{\text{adjacent}} \right) }}$

### Example

For a right triangle with hypotenuse length $25~\mathrm{feet}$ and acute angle $A^\circ$with opposite side length $12~\mathrm{feet}$, find the acute angle to the nearest degree:

Right triangle: Find the measure of angle $A$, when given the opposite side and hypotenuse.

From angle $A$, the sides opposite and hypotenuse are given.  Therefore, use the sine trigonometric function. (Soh from SohCahToa)  Write the equation and solve using the inverse key for sine.

\displaystyle{ \begin{align} \sin{A^{\circ}} &= \frac {\text{opposite}}{\text{hypotenuse}} \\ \sin{A^{\circ}} &= \frac{12}{25} \\ A^{\circ} &= \sin^{-1}{\left( \frac{12}{25} \right)} \\ A^{\circ} &= \sin^{-1}{\left( 0.48 \right)} \\ A &=29^{\circ} \end{align} }