10.8 Right Triangle Trigonometry

Pythagoras is shown writing in a book as a young man presents him with a tablet showing a diagrammatic representation of a lyre above a drawing of the sacred tetractys. — **Figure 10.138** In the lower left corner of the fresco *The School of Athens* by Raphael, the figure in white writing in the book represents Pythagoras. Alongside him, to the right, the figure with the long, light-brown hair is said to depict Archimedes.In the lower left corner of the fresco *The School of Athens* by Raphael, the figure in white writing in the book represents Pythagoras. Alongside him, to the right, the figure with the long, light-brown hair is said to depict Archimedes. (credit: modification of work “School of Athens” by Raphael (1483–1520), Vatican Museums/Wikimedia, Public Domain)

Learning Objectives

After completing this section, you should be able to:

Apply the Pythagorean Theorem to find the missing sides of a right triangle.
Apply the $30 degrees - 60 degrees - 90 degrees$ and $45 degrees - 45 degrees - 90 degrees$ right triangle relationships to find the missing sides of a triangle.
Apply trigonometric ratios to find missing parts of a right triangle.
Solve application problems involving trigonometric ratios.

This is another excerpt from Raphael’s The School of Athens. The man writing in the book represents Pythagoras, the namesake of one of the most widely used formulas in geometry, engineering, architecture, and many other fields, the Pythagorean Theorem. However, there is evidence that the theorem was known as early as 1900–1100 BC by the Babylonians. The Pythagorean Theorem is a formula used for finding the lengths of the sides of right triangles.

Born in Greece, Pythagoras lived from 569–500 BC. He initiated a cult-like group called the Pythagoreans, which was a secret society composed of mathematicians, philosophers, and musicians. Pythagoras believed that everything in the world could be explained through numbers. Besides the Pythagorean Theorem, Pythagoras and his followers are credited with the discovery of irrational numbers, the musical scale, the relationship between music and mathematics, and many other concepts that left an immeasurable influence on future mathematicians and scientists.

The focus of this section is on right triangles. We will look at how the Pythagorean Theorem is used to find the unknown sides of a right triangle, and we will also study the special triangles, those with set ratios between the lengths of sides. By ratios we mean the relationship of one side to another side. When you think about ratios, you should think about fractions. A fraction is a ratio, the ratio of the numerator to the denominator. Finally, we will preview trigonometry. We will learn about the basic trigonometric functions, sine, cosine and tangent, and how they are used to find not only unknown sides but unknown angles, as well, with little information.

Pythagorean Theorem

The Pythagorean Theorem is used to find unknown sides of right triangles. The theorem states that the sum of the squares of the two legs of a right triangle equals the square of the hypotenuse (the longest side of the right triangle).

For example, given that side $a equals 6,$ and side $b equals 8,$ we can find the measure of side $c$ using the Pythagorean Theorem. Thus,

$\begin{matrix} a^{2} + b^{2} & = & c^{2} \\ {(6)}^{2} + {(8)}^{2} & = & c^{2} \\ 36 + 64 & = & c^{2} \\ 100 & = & c^{2} \\ \sqrt{100} & = & \sqrt{c^{2}} \\ 10 & = & c \end{matrix}$

Distance

The applications of the Pythagorean Theorem are countless, but one especially useful application is that of distance. In fact, the distance formula stems directly from the theorem. It works like this:

In Figure 10.141, the problem is to find the distance between the points $(- 3, - 1)$ and $open parenthesis 3, 2 close parenthesis.$ We call the length from point $(- 3, - 1)$ to point $open parenthesis 3, minus 1 close parenthesis$ side $a$ , and the length from point $open parenthesis 3, minus 1 close parenthesis$ to point $open parenthesis 3, 2 close parenthesis$ side $b$ . To find side $c$ , we use the distance formula and we will explain it relative to the Pythagorean Theorem. The distance formula is $d = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}},$ such that $open parenthesis x sub 2 minus x sub 1 close parenthesis$ is a substitute for $a$ in the Pythagorean Theorem and is equal to $3 - (- 3) = 6;$ and $open parenthesis y sub 2 minus y sub 1 close parenthesis$ is a substitute for $b$ in the Pythagorean Theorem and is equal to $2 - (- 1) = 3.$ When we plug in these numbers to the distance formula, we have

$\begin{matrix} d & = & \sqrt{{(3 - (- 3))}^{2} + {(2 - (- 1))}^{2}} \\ = & \sqrt{{(6)}^{2} + {(3)}^{2}} = \sqrt{36 + 9} \\ = & \sqrt{45} = 3 \sqrt{5} = 6.7 \end{matrix}$

Thus, $d = c$ , the hypotenuse, in the Pythagorean Theorem.

A right triangle is plotted on an x y coordinate grid. The vertices of the triangle are (negative 3, negative 1), (3, negative 1), and (3, 2). The distance from (negative 3, negative 1) to (3, 2) is labeled c. The distance from (negative 3, negative 1) to (3, negative 1) is labeled a. The distance from (3, negative 1) to (3, 2) is labeled b. — **Figure 10.141** Distance

Example 3 Calculating Distance with the Pythagorean Theorem

The city has specific building codes for wheelchair ramps. Every vertical rise of 1 in requires that the horizontal length be 12 inches. You are constructing a ramp at your business. The plan is to make the ramp 130 inches in horizontal length and the slanted distance will measure approximately 132.4 inches (Figure 10.143). What should the vertical height be?

Figure 10.143

The Pythagorean Theorem states that the horizontal length of the base of the ramp, side a, is 130 in. The length of c, or the length of the hypotenuse, is 132.4 in. The length of the height of the triangle is side b.

Then, by the Pythagorean Theorem, we have:

$\begin{matrix} a^{2} + b^{2} & = & c^{2} \\ {(130)}^{2} + b^{2} & = & {(132.4)}^{2} \\ 16,900 + b^{2} & = & 17,529.76 \\ b^{2} & = & 17,529.76 - 16,900 \\ b^{2} & = & 629.8 \\ b & = & \sqrt{629.8} = 25 \end{matrix}$

If you construct the ramp with a 25 in vertical rise, will it fulfill the building code? If not, what will have to change?

The building code states 12 in of horizontal length for each 1 in of vertical rise. The vertical rise is 25 in, which means that the horizontal length has to be $12 open parenthesis 25 close parenthesis equals 300 in.$ So, no, this will not pass the code. If you must keep the vertical rise at 25 in, what will the other dimensions have to be? Since we need a minimum of 300 in for the horizontal length:

$\begin{matrix} {(300)}^{2} + {(25)}^{2} & = & c^{2} \\ 90,625 & = & c^{2} \\ \sqrt{90,625} & = & c = 301 in \end{matrix}$

The new ramp will look like Figure 10.144.

Figure 10.144

$30 degrees - 60 degrees - 90 degrees$ Triangles

In geometry, as in all fields of mathematics, there are always special rules for special circumstances. An example is the perfect square rule in algebra. When expanding an expression like $open parenthesis 2 x plus 5 y close parenthesis squared,$ we do not have to expand it the long way:

$\begin{matrix} {(2 x + 5 y)}^{2} & = & (2 x + 5 y) (2 x + 5 y) \\ = & {(2 x)}^{2} + 10 x y + 10 x y + {(5 y)}^{2} \\ = & 4 x^{2} + 20 x y + 25 y^{2} \end{matrix}$

If we know the perfect square formula, given as

${(a + b)}^{2} = a^{2} + 2 a b + b^{2},$

we can skip the middle step and just start writing down the answer. This may seem trivial with problems like ${(a + b)}^{2} .$ However, what if you have a problem like ${(2 \sqrt{3} + 3 \sqrt[3]{31.8 c})}^{2} ?$ That is a different story. Nevertheless, we use the same perfect square formula. The same idea applies in geometry. There are special formulas and procedures to apply in certain types of problems. What is needed is to remember the formula and remember the kind of problems that fit. Sometimes we believe that because a formula is labeled special, we will rarely have use for it. That assumption is incorrect. So, let us identify the $30 degrees - 60 degrees - 90 degrees$ triangle and find out why it is special. See Figure 10.145.

A right triangle with its legs marked x and x times square root of 3. The hypotenuse is marked 2 x. The angles at the top, bottom-left, and bottom-right are labeled 60 degrees, 90 degrees, and 30 degrees. — **Figure 10.145** The $3 0 degrees - 6 0 degrees - 9 0 degrees$

We see that the shortest side is opposite the smallest angle, and the longest side, the hypotenuse, will always be opposite the right angle. There is a set ratio of one side to another side for the $30 degrees - 60 degrees - 90 degrees$ triangle given as $1: the square root of 3: 2,$ or $x: x the square root of 3: 2 x.$ Thus, you only need to know the length of one side to find the other two sides in a $30 degrees - 60 degrees - 90 degrees$ triangle.

Example 4 Finding Missing Lengths in a $3 0 degrees - 6 0 degrees - 9 0 degrees$ Triangle

Find the measures of the missing lengths of the triangle (Figure 10.146).

Figure 10.146

We can see that this is a $30 degrees - 60 degrees - 90 degrees$ triangle because we have a right angle and a $30 degrees$ angle. The remaining angle, therefore, must equal $60 degrees.$ Because this is a special triangle, we have the ratios of the sides to help us identify the missing lengths. Side $a$ is the shortest side, as it is opposite the smallest angle $30 degrees,$ and we can substitute $a = x .$ The ratios are $x: x the square root of 3: 2 x.$ We have the hypotenuse equaling 10, which corresponds to side $c$ , and side $c$ is equal to 2 $x$ . Now, we must solve for $x$ :

$\begin{matrix} 2 x & = & 10 \\ x & = & 5 \end{matrix}$

Side $b$ is equal to $x the square root of 3$ or $5 the square root of 3.$ The lengths are $5, 5 the square root of 3, 10.$

$4 5 degrees - 4 5 degrees - 9 0 degrees$ Triangles

The $45 degrees - 45 degrees - 90 degrees$ triangle is another special triangle such that with the measure of one side we can find the measures of all the sides. The two angles adjacent to the $90 degrees$ angle are equal, and each measures $45 degrees.$ If two angles are equal, so are their opposite sides. The ratio among sides is $1: 1: the square root of 2,$ or $x: x: x the square root of 2,$ as shown in Figure 10.148.

Two right triangles. In the first triangle, the legs measure 1 and 1. The hypotenuse measures the square root of 2. The angles measure 90 degrees, 45 degrees, and 45 degrees. In the second triangle, the legs measure x and x. The hypotenuse measures x times the square root of 2. The angles measure 90 degrees, 45 degrees, and 45 degrees. — **Figure 10.148** $4 5 degrees - 4 5 degrees - 9 0 degrees$ Triangles

Trigonometry Functions

Trigonometry developed around 200 BC from a need to determine distances and to calculate the measures of angles in the fields of astronomy and surveying. Trigonometry is about the relationships (or ratios) of angle measurements to side lengths in primarily right triangles. However, trigonometry is useful in calculating missing side lengths and angles in other triangles and many applications.

Trigonometry is based on three functions. We title these functions using the following abbreviations:

$\sin = sine$
$\cos = cosine$
$\tan = tangent$

Letting $r = \sqrt{x^{2} + y^{2}},$ which is the hypotenuse of a right triangle, we have Table 10.1. The functions are given in terms of $x$ , $y$ , and $r$ , and in terms of sides relative to the angle, like opposite, adjacent, and the hypotenuse.

Table 10.1

\sin θ = \frac{y}{r} = \frac{o p p}{h y p}

\cos θ = \frac{x}{r} = \frac{a d j}{h y p}

\tan θ = \frac{y}{x} = \frac{o p p}{a d j}

We will be applying the sine function, cosine function, and tangent function to find side lengths and angle measurements for triangles we cannot solve using any of the techniques we have studied to this point. In Figure 10.150, we have an illustration mainly to identify $r$ and the sides labeled $x$ and $y$ .

Two rays are plotted on an x y coordinate plane. Both rays begin at the origin. The first ray lies on the positive x-axis. The second ray lies in the first quadrant and a point, (x, y) is marked on the ray. The angle made by the two rays is marked theta. The distance from the origin to the point along the ray is labeled r. — **Figure 10.150** Angle $θ$

An angle $θ$ sweeps out in a counterclockwise direction from the positive $x$ -axis and stops when the angle reaches the desired measurement. That ray extending from the origin that marks $θ^{\circ}$ is called the terminal side because that is where the angle terminates. Regardless of the information given in the triangle, we can find all missing sides and angles using the trigonometric functions. For example, in Figure 10.151, we will solve for the missing sides.

Two rays are plotted on an x y coordinate plane. The ray lies in the first quadrant and a point is marked on the ray. A vertical line is drawn from the point to meet the x-axis and it measures y. The horizontal distance from the origin to the line is marked x. The angle made by the ray with the x-axis is marked 60 degrees. The distance from the origin to the point along the ray is labeled r equals 2. — **Figure 10.151** Solving for Missing Sides

Let’s use the trigonometric functions to find the sides $x$ and $y$ . As long as your calculator mode is set to degrees, you do not have to enter the degree symbol. First, let’s solve for $y$ .

We have $\sin θ = \frac{y}{r},$ and $θ equals 60 degrees.$ Then,

$\begin{matrix} \sin 60^{\circ} & = & \frac{y}{2} \\ 2 \sin 60^{\circ} & = & y \\ 1.732 & = & y \\ \sqrt{3} & = & y \end{matrix}$

Next, let’s find $x$ . This is the cosine function. We have $\cos θ = \frac{x}{r} .$ Then,

$\begin{matrix} \cos 60^{\circ} & = & \frac{x}{2} \\ 2 \cos 60^{\circ} & = & x \\ = & 1 \end{matrix}$

Now we have all sides, $1, the square root of 3, 2.$ You can also check the sides using the $30 degrees - 60 degrees - 90 degrees$ ratio of $1: the square root of 3: 2.$ Table 10.2 is a list of common angles, which you should find helpful.

**Table 10.2**
$sin 0 degrees equals 0$	$cos 0 degrees equals 1$
$\sin 30^{\circ} = \frac{1}{2}$	$\cos 30^{\circ} = \frac{\sqrt{3}}{2}$
$\sin 45^{\circ} = \frac{\sqrt{2}}{2}$	$\cos 45^{\circ} = \frac{\sqrt{2}}{2}$
$\sin 60^{\circ} = \frac{\sqrt{3}}{2}$	$\cos 60^{\circ} = \frac{1}{2}$
$sin 90 degrees equals 1$	$cos 90 degrees equals 0$

Example 7 Using Trigonometric Functions

Find the lengths of the missing sides for the triangle (Figure 10.152).

Figure 10.152

We have a $55 degrees$ angle, and the length of the triangle on the $x$ -axis is 6 units.

Step 1: To find the length of $r$ , we can use the cosine function, as $\cos θ = \frac{x}{r} .$ We manipulate this equation a bit to solve for $r$ :

$\begin{matrix} \cos (55^{\circ}) & = & \frac{6}{r} \\ r \cos (55^{\circ}) & = & 6 \\ r & = & \frac{6}{\cos (55^{\circ})} \\ r & = & \frac{6}{0.5736} = 10.46 \end{matrix}$

Step 2: We can use the Pythagorean Theorem to find the length of $y$ . Prove that your answers are correct by using other trigonometric ratios:

$\begin{matrix} 6^{2} + y^{2} & = & {10.46}^{2} \\ y^{2} & = & 109.4 - 36 \\ y & = & 8.57 \end{matrix}$

Step 3: Now that we have $y$ , we can use the sine function to prove that $r$ is correct. We have $\sin θ = \frac{y}{r} .$

$\begin{matrix} \sin (55^{\circ}) & = & \frac{8.57}{r} \\ r \sin (55^{\circ}) & = & 8.57 \\ r & = & \frac{8.57}{\sin (55^{\circ})} \\ = & \frac{8.57}{0.819} = 10.46 \end{matrix}$

To find angle measurements when we have two side measurements, we use the inverse trigonometric functions symbolized as $\sin^{- 1},$ $\cos^{- 1},$ or $\tan^{- 1} .$ The –1 looks like an exponent, but it means inverse. For example, in the previous example, we had $x equals 6$ and $r equals 10.46.$ To find what angle has these values, enter the values for the inverse cosine function $\cos^{- 1} (\frac{x}{r})$ in your calculator:

$\cos^{- 1} (\frac{6}{10.46}) = 55^{\circ} .$

You can also use the inverse sine function and enter the values of $\sin^{- 1} (\frac{y}{r})$ in your calculator given $y equals 8.57$ and $r equals 10.46.$ We have

$\sin^{- 1} (\frac{8.57}{10.46}) = 55^{\circ} .$

Finally, we can also use the inverse tangent function. Recall $\tan θ = \frac{y}{x} .$ We have

$\tan^{- 1} (\frac{8.57}{6}) = 55^{\circ} .$

Angle of Elevation and Angle of Depression

Other problems that involve trigonometric functions include calculating the angle of elevation and the angle of depression. These are very common applications in everyday life. The angle of elevation is the angle formed by a horizontal line and the line of sight from an observer to some object at a higher level. The angle of depression is the angle formed by a horizontal line and the line of sight from an observer to an object at a lower level.

People in Mathematics

Pythagoras and the Pythagoreans

The Pythagorean Theorem is so widely used that most people assume that Pythagoras (570–490 BC) discovered it. The philosopher and mathematician uncovered evidence of the right triangle concepts in the teachings of the Babylonians dating around 1900 BC. However, it was Pythagoras who found countless applications of the theorem leading to advances in geometry, architecture, astronomy, and engineering.

Among his accolades, Pythagoras founded a school for the study of mathematics and music. Students were called the Pythagoreans, and the school’s teachings could be classified as a religious indoctrination just as much as an academic experience. Pythagoras believed that spirituality and science coexist, that the intellectual mind is superior to the senses, and that intuition should be honored over observation.

Pythagoras was convinced that the universe could be defined by numbers, and that the natural world was based on mathematics. His primary belief was All is Number. He even attributed certain qualities to certain numbers, such as the number 8 represented justice and the number 7 represented wisdom. There was a quasi-mythology that surrounded Pythagoras. His followers thought that he was more of a spiritual being, a sort of mystic that was all-knowing and could travel through time and space. Some believed that Pythagoras had mystical powers, although these beliefs were never substantiated.

Pythagoras and his followers contributed more ideas to the field of mathematics, music, and astronomy besides the Pythagorean Theorem. The Pythagoreans are credited with the discovery of irrational numbers and of proving that the morning star was the planet Venus and not a star at all. They are also credited with the discovery of the musical scale and that different strings made different sounds based on their length. Some other concepts attributed to the Pythagoreans include the properties relating to triangles other than the right triangle, one of which is that the sum of the interior angles of a triangle equals $180 degrees.$ These geometric principles, proposed by the Pythagoreans, were proven 200 years later by Euclid.

Who Knew?

A Visualization of the Pythagorean Theorem

In Figure 10.158, which is one of the more popular visualizations of the Pythagorean Theorem, we see that square $a$ is attached to side $a$ ; square $b$ is attached to side $b$ ; and the largest square, square $c$ , is attached to side $c$ . Side $a$ measures 3 cm in length, side $b$ measures 4 cm in length, and side $c$ measures 5 cm in length. By definition, the area of square $a$ measures 9 square units, the area of square $b$ measures 16 square units, and the area of square $c$ measures 25 square units. Substitute the values given for the areas of the three squares into the Pythagorean Theorem and we have

$\begin{matrix} a^{2} + b^{2} & = & c^{2} \\ 3^{2} + 4^{2} & = & 5^{2} \\ 9 + 16 & = & 25 \end{matrix}$

Thus, the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse, as stated in the Pythagorean Theorem.

Figure 10.158

Key Terms

right triangle
sine
cosine
tangent

Key Concepts

The Pythagorean Theorem is applied to right triangles and is used to find the measure of the legs and the hypotenuse according the formula $a^{2} + b^{2} = c^{2},$ where c is the hypotenuse.
To find the measure of the sides of a special angle, such as a $30 degrees - 60 degrees - 90 degrees$ triangle, use the ratio $x: x the square root of 3: 2 x,$ where each of the three sides is associated with the opposite angle and 2 $x$ is associated with the hypotenuse, opposite the $90 degrees$ angle.
To find the measure of the sides of the second special triangle, the $45 degrees - 45 degrees - 90 degrees$ triangle, use the ratio $x: x: x the square root of 2,$ where each of the three sides is associated with the opposite angle and $x the square root of 2$ is associated with the hypotenuse, opposite the $90 degrees$ angle.
The primary trigonometric functions are $\sin θ = \frac{o p p}{h y p},$ $\cos θ = \frac{a d j}{h y p},$ and $\tan θ = \frac{o p p}{a d j} .$
Trigonometric functions can be used to find either the length of a side or the measure of an angle in a right triangle, and in applications such as the angle of elevation or the angle of depression formed using right triangles.

Formula

The Pythagorean Theorem states

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

where $a$ and $b$ are two sides (legs) of a right triangle and $c$ is the hypotenuse.

Projects

One of the reasons so many formulas in geometry were discovered was because of the importance in finding measurements of lengths, areas, perimeter, and angles. Find at least five examples of how geometry can be used in practical applications today.
Who were the Pythagoreans? Why did this society exist? Explore what they did and discuss some of their beliefs.

Adapted from Contemporary Mathematics by OpenStax (openstax.org), licensed under CC BY-NC-SA 4.0. Changes were made. License: CC-BY-NC-SA-4.0.