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# Applications of Cosine

## Use a calculator to find cosine of given angle

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Determine and Use the Cosine Ratio

Arnold needs to open the door into the hayloft of the barn. He leans the 20-foot ladder against the outside wall and uses a measuring device to determine the size of the angle the ladder makes with the ground.

Arnold is happy because the angle of \begin{align*}75^\circ\end{align*} means it is safe to climb the ladder. The ladder must be left in place for everyone to get in and out of the loft. If the ladder is at least 5 feet from the barn, Arnold can hook up the portable food and drink station.

How can Arnold figure out the distance from the base of the barn to the foot of the ladder before the portable station arrives?

In this concept, you will learn to determine and use the cosine ratio.

### Cosine Ratio

The TI-calculator can be used to find the ratio when the measure of the angle is known. If the measure of \begin{align*} \angle B= 68^ \circ\end{align*} then the value of the cosine ratio can be found by following the Key Press History:

On the calculator screen the following is displayed:

This decimal should always be rounded to the nearest ten thousandth (four places after the decimal).

\begin{align*}\cos 68^ \circ = 0.3746\end{align*}

This cosine ratio for an angle of  \begin{align*}68 ^\circ\end{align*} will remain constant regardless of the size of the triangle. The sides will always be in the same proportion to each other. The value of the Cosine ratio will always be between zero and one.

The Cosine ratio is related to one of the acute angles of a right triangle such that the cosine of the acute angle is the ratio of the side adjacent to the specific acute angle (the reference angle) to the hypotenuse.

The Cosine ratio can be written as  \begin{align*}\cos \angle = \frac{\text {adjacent}}{\text {hypotenuse}}\end{align*}. This equation has three parts to it – an angle and two sides. When the lengths of the two sides are known, the measure of the angle can be calculated. When the measure of an angle and the length of one side is known, the length of the other side can be calculated.

It is the sides of the triangle that determine the trigonometric ratio that will be used to calculate the measure of an angle or the length of a side. To use the cosine ratio, the adjacent side and the hypotenuse of the right triangle must be indicated. These two sides will have values on them if the measure of an angle is to be calculated using the cosine ratio. If the length of a side is to be calculated using the cosine ratio then one of the sides will display a value and the other will display a variable (the side that is unknown).

Let’s look at the following right triangle to see how this works.

The measure of \begin{align*} \angle A \text{ is } 52^ \circ\end{align*}. The length of the hypotenuse is \begin{align*}18 \text{ centimeters}\end{align*}. The side \begin{align*}\overline{AC}\end{align*} has the variable ‘\begin{align*}X\end{align*}’ on it which means this is the side that is unknown and its length must be calculated.

The sides of the triangle can be named using the acute angle B which is the reference angle for this triangle.

The two sides that are indicated on this triangle are the hypotenuse and the adjacent. The Cosine ratio is the ratio of the adjacent side to the hypotenuse.

First, write the cosine ratio using words.

\begin{align*}\cos A = \frac{\text{adjacent}}{\text{hypotenuse}}\end{align*}

Next, write the cosine ratio using symbols.

\begin{align*}\cos A = \frac{AC}{AB}\end{align*}

Next, fill all known values into the equation. \begin{align*}\angle A = 52^\circ; \ \overline{AC} = X; \ \overline{AB} = 18\end{align*}.

\begin{align*}\cos 52^\circ = \frac{X}{18}\end{align*}

Next, use the TI-calculator to find the value of \begin{align*} \cos 52^\circ\end{align*}. Round the decimal to the nearest ten thousandth.

\begin{align*}\cos 52^\circ = 0.6157\end{align*}

Next, substitute this value into the equation.

\begin{align*}\begin{array}{rcl} \cos 52^\circ &=& \frac{X}{18} \\ 0.6157 &=& \frac{X}{18} \\ \end{array}\end{align*}

Next, multiply both sides of the equation by 18 to solve for the variable.

\begin{align*}\begin{array}{rcl} 0.6157 &=& \frac{X}{18} \\ 18(0.6157) &=& 18 \left (\frac{X}{18} \right) \\ 11.0826 &=& { \overset{1} {\cancel{18}} } \left ( \frac{X}{ \cancel{18}} \right) \\ 11.08 &=& X \\ \end{array}\end{align*}

The length of the adjacent side of the right triangle is 11.08 centimeters.

The calculated lengths of sides are usually expressed to the nearest hundredth while the calculated measure of an angle is usually expressed to the nearest tenth, unless otherwise stated.

### Examples

#### Example 1

Earlier, you were given a problem about Arnold and the hayloft door. He needs to know if there is enough room between the barn and the ladder for the portable station. How can he figure this out?

He can use the cosine ratio.

First, draw and label a right triangle to model the problem.

Next, name the sides of the triangle using the reference \begin{align*}\angle A\end{align*}.

Next, write the cosine ratio using words.

\begin{align*}\cos A = \frac{\text{ adjacent }}{\text { hypotenuse }}\end{align*}

Next, write the cosine ratio using symbols.

\begin{align*}\cos 75 = \frac{AC}{AB}\end{align*}

Next, fill all known values into the equation. \begin{align*}\angle A = 75^\circ; \ \overline{AC} = X; \ \overline{AB} =20 \ ft\end{align*}

\begin{align*}\cos75 = \frac{X}{20}\end{align*}

Next, use the TI-calculator to find the value of \begin{align*}\cos75^\circ\end{align*}.

\begin{align*}\cos75^\circ = 0.2588\end{align*}

Next, substitute this value into the equation.

\begin{align*}\begin{array}{rcl} \cos75^\circ &=& \frac{X}{20} \\ 0.2588 &=& \frac{X}{20} \\ \end{array}\end{align*}

Next, multiply both sides of the equation by 20 to solve for the variable.

\begin{align*}\begin{array}{rcl} 0.2588 &=& \frac{X}{20} \\ 20(0.2588) &=& 20 \left (\frac{X}{20} \right) \\ 5.176 &=& { \overset{1} {\cancel{20}} } \left ( \frac{X}{ \cancel{20}} \right) \\ 5.18 &=& X \end{array}\end{align*}

The distance is 5.18 feet, so Arnold can put the portable station in place.

#### Example 2

For the following right triangle calculate the length of side ‘\begin{align*}X\end{align*}’ to the nearest hundredth.

First, use the reference angle \begin{align*}D\end{align*} to name the sides of the triangle.

The two sides that are indicated on this triangle are the adjacent and the hypotenuse. The Cosine ratio is the ratio of the adjacent side to the hypotenuse.

First, write the cosine ratio using words.

\begin{align*}\cos D = \frac{\text{adjacent}}{\text{hypotenuse}}\end{align*}

Next, write the cosine ratio using symbols.

\begin{align*}\cos D = \frac{DE}{DF}\end{align*}

Next, fill all known values into the equation. \begin{align*}\angle D = 35^\circ; \ \overline{DF} = X; \ \overline{DE} = 8.2\end{align*}.

\begin{align*}\begin{array}{rcl} \cos D &=& \frac{DE}{DF} \\ \cos 35^\circ &=& \frac{8.2}{X} \end{array}\end{align*}

Next, use the TI-calculator to find the value of \begin{align*}\cos35^\circ\end{align*}.

\begin{align*}\cos35^\circ = 0.8192\end{align*}

Next, substitute this value into the equation.

\begin{align*}\begin{array}{rcl} \cos35^\circ &=& \frac{8.2}{X} \\ 0.8192 &=& \frac{8.2}{X} \end{array}\end{align*}

Next, multiply both sides of the equation by \begin{align*}X\end{align*} to clear the denominator.

\begin{align*}\begin{array}{rcl} 0.8192 &=& \frac{8.2}{X} \\ X(0.8192) &=& X \left (\frac{8.2}{X} \right) \\ 0.8192 \ X &=& { \overset{1} {\cancel{X}} } \left ( \frac{8.2}{ \cancel{X}} \right) \\ 0.8192 \ X &=& 8.2 \\ \end{array}\end{align*}

Then, divide both sides of the equation by 0.8192 to solve for the variable.

\begin{align*}\begin{array}{rcl} 0.8192X &=& 8.2 \\ \frac{\overset{1}{\cancel{0.8192X}}}{\cancel{0.8192}} &=& \frac{8.2}{0.8192} \\ X &=& 10.01 \\ \end{array}\end{align*}

The length of the hypotenuse of the right triangle is 10.01 inches.

#### Example 3

For the following right triangle calculate the length of side ‘\begin{align*}X\end{align*}’.

First, use the reference angle \begin{align*}C\end{align*}  to name the sides of the triangle.

The two sides that are indicated on this triangle are the adjacent and the hypotenuse. The Cosine ratio is the ratio of the adjacent side to the hypotenuse.

First, write the cosine ratio using words.

\begin{align*}\cos C = \frac{\text{adjacent}}{\text{hypotenuse}}\end{align*}

Next, write the cosine ratio using symbols.

\begin{align*}\cos C = \frac{BC}{AC}\end{align*}

Next, fill all known values into the equation. \begin{align*}\angle C = 59^\circ; \ \overline{BC} = X; \ \overline{AC} =38\end{align*}.

\begin{align*}\begin{array}{rcl0} \cos C &=& \frac{BC}{AC} \\ \cos 59^\circ &=& \frac{X}{38} \\ \end{array}\end{align*}

Next, use the TI-calculator to find the value of \begin{align*}\cos59^\circ\end{align*}.

\begin{align*}\cos59^\circ = 0.5150\end{align*}

Next, substitute this value into the equation.

\begin{align*}\begin{array}{rcl} \cos 59^\circ &=& \frac{X}{38} \\ 0.5150 &=& \frac{X}{38} \\ \end{array}\end{align*}

Next, multiply both sides of the equation by 38 to solve for the variable.

\begin{align*}\begin{array}{rcl} 0.5150 &=& \frac{X}{38} \\ 38(0.5150) &=& 38 \left (\frac{X}{38} \right) \\ 38(0.5150) &=& { \overset{1} {\cancel{38}} } \left ( \frac{X}{ \cancel{38}} \right) \\ 19.57 &=& X \\ \end{array}\end{align*}

The length of the adjacent side of the right triangle is 19.57 feet.

#### Example 4

Answer the questions below for the following right triangle.

1. What angle is the reference angle of \begin{align*}\triangle JKL\end{align*} ?
2. What is the measure of the reference angle?
3. What two sides of the triangle are indicated in \begin{align*}\triangle JKL\end{align*} ?
4. Write the trigonometric ratio in words that would be used to calculate the length of \begin{align*}\overline{JL}\end{align*} ?
5. What is the value of \begin{align*} \cos40^\circ\end{align*}?

The answers to the above questions for \begin{align*}\triangle JKL\end{align*} are:

1. The acute angle referred to in \begin{align*}\triangle JKL\end{align*} is \begin{align*}\angle J\end{align*}.
2. The measure of the reference angle is \begin{align*}40^\circ\end{align*}.
3. The two sides of \begin{align*}\triangle JKL\end{align*} that are indicated are  \begin{align*}\overline{JK}\end{align*} the adjacent side and  \begin{align*}\overline{JL}\end{align*} the hypotenuse.
4. \begin{align*}\cos J = \frac{\text{ adjacent }}{\text{ hypotenuse }}\end{align*}
5. The value of \begin{align*} \cos40^\circ \text { is } 0.7660.\end{align*}

### Review

Use a calculator to find each Cosine.You may round to the nearest hundredth.

1. \begin{align*} \text{Cosine} \ 33^{\circ}\end{align*}

2. \begin{align*}\text{Cosine} \ 29^{\circ}\end{align*}

3. \begin{align*}\text{Cosine} \ 73^{\circ}\end{align*}

4. \begin{align*}\text{Cosine} \ 88^{\circ}\end{align*}

5. \begin{align*}\text{Cosine} \ 50^{\circ}\end{align*}

6. \begin{align*}\text{Cosine} \ 67^{\circ}\end{align*}

7. \begin{align*}\text{Cosine} \ 42^{\circ}\end{align*}

8. \begin{align*}\text{Cosine} \ 18^{\circ}\end{align*}

9. \begin{align*}\text{Cosine} \ 9^{\circ}\end{align*}

Find the length of the adjacent side.

10. A triangle has a hypotenuse of 7 inches. Angle A is equal to 60 degrees. Find the length of the adjacent side.

11. A triangle has a hypotenuse of 12 inches. Angle B is equal to 45 degrees. Find the length of the adjacent side.

12. A triangle has a hypotenuse of 8 inches. Angle A is equal to 35 degrees. Find the length of the adjacent side.

13. A triangle has a hypotenuse of 12 inches. Angle A is equal to 28 degrees. Find the length of the adjacent side.

14. A triangle has a hypotenuse of 6 inches. Angle A is equal to 33 degrees. Find the length of the adjacent side.

15. A triangle has a hypotenuse of 14 inches. Angle A is equal to 72 degrees. Find the length of the adjacent side.

16. A triangle has a hypotenuse of 11 inches. Angle A is equal to 80 degrees. Find the length of the adjacent side.

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### Vocabulary Language: English

cosine

The cosine of an angle in a right triangle is a value found by dividing the length of the side adjacent the given angle by the length of the hypotenuse.

sine

The sine of an angle in a right triangle is a value found by dividing the length of the side opposite the given angle by the length of the hypotenuse.

Tangent

The tangent of an angle in a right triangle is a value found by dividing the length of the side opposite the given angle by the length of the side adjacent to the given angle.

Trigonometric Ratios

Ratios that help us to understand the relationships between sides and angles of right triangles.

1. [1]^ License: CC BY-NC 3.0
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16. [16]^ License: CC BY-NC 3.0
17. [17]^ License: CC BY-NC 3.0

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