Lois, who is enjoying her lunch in the park, notices a group of students from the University Forestry class measuring the length of a shadow cast by a large spruce tree. The students are also measuring the angle made by the sun with the ground.

The shadow is 12 meters long and the angle of the sun is 46°. How can the students calculate the height of the tree without actually measuring it?

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

### Tangent Ratios

You have used the TI-calculator to determine the measure of an angle using the inverse tangent function \begin{align*}\left(\tan^{-1} \right)\end{align*} when the value of the ratio was known. If \begin{align*}\tan B = 3.1789\end{align*} then using the TI-calculator displayed the measure of 72.5° for the angle.

The TI-calculator can also be used to find the ratio when the measure of the angle is known. If the measure of \begin{align*}\angle B = 37^\circ\end{align*} then the value of the tangent 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*}\tan 37^\circ = 0.7536\end{align*}

This tangent ratio for an angle of 37° will remain constant regardless of the size of the triangle. The sides will always be in the same proportion to each other. The Tangent ratio can have very large or very small values depending on the triangle.

The Tangent ratio is related to one of the acute angles of a right triangle such that the tangent of the acute angle is the ratio of the side adjacent to the specific acute angle (the reference angle) to the side opposite the acute angle. The Tangent ratio can be written as \begin{align*}\tan \angle = \frac{\text{opposite}}{\text{adjacent}}\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 tangent ratio, the adjacent and opposite sides of the right triangle, in relation to the reference angle, must be indicated. These two sides will have values on them if the measure of an angle is to be calculated using the tangent ratio. If the length of a side is to be calculated using the tangent 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\end{align*} is 37°. The length of side \begin{align*}\overline{AC}\end{align*} is 21 inches. The side \begin{align*}\overline{BC}\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

which is the reference angle for this triangle.

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

First, write the tangent ratio using words.

\begin{align*}\tan A = \frac{\text{opposite}}{\text{adjacent}}\end{align*}

Next, write the tangent ratio using symbols.

Next, fill all known values into the equation. \begin{align*}\angle A = 37^\circ; \ \overline{AC} = 21; \ \overline{BC} = X\end{align*}.

\begin{align*}\tan 37^\circ = \frac{X}{21}\end{align*}

Next, use the TI-calculator to find the value of 37°. Round the decimal to the nearest ten thousandth.

\begin{align*}\tan 37^\circ = 0.7536\end{align*}

Next, substitute this value into the equation.

\begin{align*}\begin{array}{rcl} \tan 37^\circ &=& \frac{X}{21}\\ \\ 0.7536 &=& \frac{X}{21} \end{array}\end{align*}

Next, multiply both sides of the equation by 21 to solve for the variable.

\begin{align*}\begin{array}{rcl} 0.7536 &=& \frac{X}{21}\\ \\ 21(0.7536) &=& 21 \left(\frac{X}{21} \right)\\ \\ 15.8256 &=& \overset{{1}}{\cancel{21}} \left(\frac{X}{{\cancel{21}}} \right)\\ \\ 15.83 &=& X \end{array}\end{align*}

The answer is 15.83

The length of the opposite side of the right triangle is 15.83 inches.

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 the students and the spruce tree. They need to calculate the height of the spruce tree without actually measuring it.

To do this, they can use the tangent ratio.

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

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

Next, write the Tangent ratio using words.

\begin{align*}\tan A = \frac{\text{opposite}}{\text{adjacent}}\end{align*}

Next, write the Tangent ratio using symbols.

\begin{align*}\tan A = \frac{BC}{AC}\end{align*}

Next, fill all known values into the equation. \begin{align*}\angle A = 46^\circ; \ \overline{AC} = 12; \ \overline{BC} = X\end{align*}.

\begin{align*}\tan 46^\circ = \frac{X}{12}\end{align*}

Next, use the TI-calculator to find the value of \begin{align*}\tan 46^\circ\end{align*}.

\begin{align*}\tan 46^\circ = 1.0355\end{align*}

Next, substitute this value into the equation.

\begin{align*}\begin{array}{rcl} \tan 46^\circ &=& \frac{X}{12}\\ \\ 1.0355 &=& \frac{X}{12} \end{array}\end{align*}

Next, multiply both sides of the equation by 12 to solve for the variable.

\begin{align*}\begin{array}{rcl} 1.0355 &=& \frac{X}{12}\\ 12(1.0355) &=& 12 \left(\frac{X}{12} \right)\\ 12.426 &=& \overset{{1}}{\cancel{12}} \left(\frac{X}{{\cancel{12}}} \right)\\ 12.43 &=& X \end{array}\end{align*}

The answer is 12.43.

The height of the tree is 12.43 meters.

#### 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

to name the sides of the triangle.

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

First, write the tangent ratio using words.

\begin{align*}\tan B = \frac{\text{opposite}}{\text{adjacent}}\end{align*}

Next, write the tangent ratio using symbols.

\begin{align*}\tan B = \frac{AC}{BC}\end{align*}

Next, fill all known values into the equation. \begin{align*}\angle B = 50^\circ; \ \overline{BC} = X; \ \overline{AC} = 42\end{align*}.

\begin{align*}\begin{array}{rcl} \tan B &=& \frac{AC}{BC}\\ \\ \tan 50^\circ &=& \frac{42}{X} \end{array}\end{align*}

Next, use the TI-calculator to find the value of \begin{align*}\tan 50^\circ\end{align*}.

\begin{align*}\tan 50^\circ = 1.1918\end{align*}

Next, substitute this value into the equation.

\begin{align*}\begin{array}{rcl} \tan 50^\circ &=& \frac{42}{X}\\ \\ 1.1918 &=& \frac{42}{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} 1.1918 &=& \frac{42}{X}\\ \\ X(1.1918) &=& X \left(\frac{42}{X} \right)\\ \\ 1.1918 X &=& \overset{{1}}{\cancel{X}} \left(\frac{42}{\cancel{X}} \right)\\ \\ 1.1918 X &=& 42 \end{array}\end{align*}

Then, divide both sides of the equation by 1.1918 to solve for the variable.

\begin{align*}\begin{array}{rcl} 1.1918 X &=& 42\\ \\ \frac{\overset{{1}} {\cancel{1.1918}}X}{\cancel{1.1918}} &=& \frac{42}{1.1918}\\ \\ X &=& 35.24 \end{array}\end{align*}

The answer is 35.24.

The length of the adjacent side of the right tringle is of the right triangle is 35.24 feet.

#### Example 3

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

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

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

First, write the Tangent ratio using words.

\begin{align*}\tan E = \frac{\text{opposite}}{adjacent}\end{align*}

Next, write the tangent ratio using symbols.

\begin{align*}\tan E = \frac{DF}{DE}\end{align*}

Next, fill all known values into the equation. \begin{align*}\angle E = 56^\circ; \ \overline{DF} = X; \ \overline{DE} = 24\end{align*}.

\begin{align*}\begin{array}{rcl} \tan E &=& \frac{DF}{DE}\\ \\ \tan 56^\circ &=& \frac{X}{24} \end{array}\end{align*}

Next, use the TI-calculator to find the value of \begin{align*}\tan 56^\circ\end{align*}.

\begin{align*}\tan 56^\circ = 1.4826\end{align*}

Next, substitute this value into the equation.

\begin{align*}\begin{array}{rcl} \tan 56^\circ &=& \frac{X}{24}\\ \\ 1.4826 &=& \frac{X}{24} \end{array}\end{align*}

Next, multiply both sides of the equation by 24 to solve for the variable.

\begin{align*}\begin{array}{rcl} 1.4826 &=& \frac{X}{24}\\ 24(1.4826) &=& 24 \left(\frac{X}{24} \right)\\ 24(1.4826) &=& \overset{{1}}{\cancel{24}} \left(\frac{X}{\cancel{24}} \right)\\ 35.58 &=& X \end{array}\end{align*}

The answer is 35.58.

The length of the opposite side of the right triangle is 35.58 meters.

#### Example 4

Use the TI-calculator to determine the missing measure.

i) \begin{align*}\tan 45^\circ = \frac{x}{15}\end{align*}

ii) \begin{align*}\tan B = \frac{12}{20}\end{align*}

iii) \begin{align*}\text{If } \angle A = 82^\circ \text{ then } \tan A = \underline{\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

\begin{align*}\tan 45^\circ = \frac{x}{15}\end{align*}

i) First, use the TI-calculator to find the value of \begin{align*}\tan 45^\circ\end{align*}.

\begin{align*}\tan 45^\circ = 1\end{align*}

Next, fill this value into the equation.

\begin{align*}\begin{array}{rcl} \tan 45^\circ &=& \frac{x}{15}\\ \\ 1 &=& \frac{x}{15} \end{array}\end{align*}

Then, multiply both sides of the equation by 15 to solve for the variable.

\begin{align*}\begin{array}{rcl} 1 &=& \frac{x}{15}\\ 15(1) &=& 15 \left(\frac{x}{15} \right)\\ 15 &=& {\cancel{15}} \left(\frac{x}{{\cancel {15}}} \right)\\ 15 &=& x \end{array}\end{align*}

The answer is 15.

ii) \begin{align*}\tan B = \frac{12}{20}\end{align*}

First, use the TI-calculator to express the fraction as a decimal.

\begin{align*}\tan B = \frac{12}{20}\end{align*}

Next, rewrite the equation using the decimal form of the fraction.

\begin{align*}\tan B = 0.6\end{align*}

Next, use the inverse tangent function \begin{align*}\left(\tan^{-1} \right)\end{align*} to calculate the measure of \begin{align*}\angle B\end{align*}.

Then, round the answer to the nearest tenth.

\begin{align*}\angle B = 31.0^\circ\end{align*}

iii) \begin{align*}\text{if } \angle A = 82^\circ \text{ then } \tan A = \underline{\;\;\;\;\;\;\;\;}\end{align*}

First, use the TI-calculator to determine the value of 82°.

Then, round the answer to the nearest ten thousandth.

\begin{align*}\tan 82^\circ = 7.1154\end{align*}

### Review

Use a calculator to figure out each Tangent. You may round to the nearest hundredth.

1. Tangent 7°

2. Tangent 41°

3. Tangent 65°

4. Tangent 22°

5. Tangent 18°

6. Tangent 35°

7. Tangent 50°

8. Tangent 54°

9. Tangent 66°

10. Tangent 70°

Find the length of the opposite side in each example.

11. We have a right triangle. Angle

is 45 degrees. The side of the triangle adjacent to angle is 6 inches. What is the length of the opposite side?12. We have a right triangle. Angle

is 63 degrees. The side of the triangle adjacent to angle is 7 inches. What is the length of the opposite side?13. We have a right triangle. Angle

is 29 degrees. The side of the triangle adjacent to angle is 6 inches. What is the length of the opposite side?14. We have a right triangle. Angle

is 12 degrees. The side of the triangle adjacent to angle is 4.5 inches. What is the length of the opposite side?15. We have a right triangle. Angle \begin{align*}A\end{align*} is 8 inches. What is the length of the opposite side?

is 9 degrees. The side of the triangle adjacent to angle### Review (Answers)

To see the Review answers, open this PDF file and look for section 7.18.