Using the Tangent Ratio with a Calculator Examples 2 - 4

Lesson 1: The Tangent Ratio: Using the Tangent Ratio with a Calculator Examples 2 - 4

EXAMPLE 2

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Determine the length of side s to the nearest hundredth.

Solution

Step 1: Identify and label the sides as being adjacent to, opposite or the hypotenuse in relation to the angle indicated.



Step 2: State the appropriate ratio.

\(\text{tangent of angle}\,\theta=\frac{\text{length opposite}\,\theta}{\text{length adjacent to}\,\theta}\)

Step 3: Substitute known values and calculate the unknown value.

\(\begin{align} \text{tan}\,\theta&=\frac{\text{opp}}{\text{adj}} \\ \\ \text{tan}\,33°&=\frac{6}{s} \\ \\ {\color{red}{s}}\times \text{tan}\,33°&=\frac{6}{\cancel{s}}\times \cancel{\color{red}{s}} \\ \\ s\times \text{tan}\,33°&=6 \\ \\ \frac{s\times \cancel{\text{tan}\,33°}}{\cancel{\color{red}{\text{tan}\,33°}}}&=\frac{6}{\color{red}{\text{tan}\,33°}} \\ \\ s&=9.2 \\ \end{align}\)

The length of side s is approximately 9.2 units long.

EXAMPLE 3

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Determine the value of θ to the nearest degree.

Solution

Step 1: Identify and label the sides as being adjacent to, opposite or the hypotenuse in relation to the angle indicated.



Step 2: State the appropriate ratio.

\(\text{tangent of angle}\,\theta=\frac{\text{length opposite}\,\theta}{\text{length adjacent to}\,\theta}\)

Step 3: Substitute known values and calculate the unknown value.

\(\begin{align} \text{tan}\,\theta&=\frac{\text{opp}}{\text{adj}} \\ \\ \text{tan}\,\theta&=\frac{15}{18} \\ \\ \theta&=\text{tan}^{-1}\left(\frac{15}{18}\right) \\ \theta&=40° \\ \end{align}\)

The measure of angle θ is approximately 40°.

   Multimedia

A video describing the use of tangent to solve a problem is provided.

EXAMPLE 4

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If the flagpole casts a shadow that is 5.7 m long, how tall is the flagpole? Express your answer to the nearest tenth.

Solution

Step 1: Identify and label the sides as being adjacent, opposite or the hypotenuse in relation to the angle indicated.



Step 2: State the appropriate ratio.

\(\text{tangent of angle}\,\theta=\frac{\text{length opposite}\,\theta}{\text{length adjacent to}\,\theta}\)

Step 3: Substitute known values and calculate for the unknown value.

\(\begin{align} \text{tan}\,\theta&=\frac{\text{opp}}{\text{adj}} \\ \\ \text{tan}\,66°&=\frac{h}{5.7\,\text{m}} \\ \\ {\color{red}{5.7\,\text{m}}}\times \text{tan}\,66°&=\frac{h}{\cancel{5.7\,\text{m}}}\times \cancel{\color{red}{5.7\,\text{m}}} \\ \\ 12.8\,\text{m}&=h \\ \end{align}\)

The flagpole is approximately 12.8 m tall.




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