Lesson 1: The Tangent Ratio: Solving Problems Involving the Tangent Ratio with a Table Examples


A video demonstration of the solution for Example 1 has been provided.


EXAMPLE 1

Determine the value of t.



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.

\(\begin{align} \text{tan}\,t&=\frac{\text{opp}}{\text{adj}} \\ \\ &=\frac{9.5}{5.5} \\ \\ &=1.73 \\ \end{align}\)

Looking at the table, it can be seen that tan 60° = 1.73, so t = 60°.

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A video demonstration of using tangent has been provided.


EXAMPLE 2

Determine the value of y, to the nearest tenth.



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.

\(\begin{align} \text{tan}\,\theta&=\frac{\text{opp}}{\text{adj}} \\ \\ \text{tan}\,55°&=\frac{y}{12} \\ \\ 1.43&=\frac{y}{12} \\ \\ 1.43\times {\color{red}{12}}&=\frac{y}{\cancel{12}}\times \cancel{\color{red}{12}} \\ \\ 17.2&=y \\ \end{align}\)

The value of y is approximately 17.2

EXAMPLE 3

Use the tangent table to determine the unknown side length, to the nearest tenth.



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.

\(\begin{align} \text{tan}\,\theta&=\frac{\text{opp}}{\text{adj}} \\ \\ \text{tan}\,20°&=\frac{15}{q} \\ \\ 0.36&=\frac{15}{q} \\ \\ 0.36\times {\color{red}{q}}&=\frac{15}{\cancel{q}}\times \cancel{\color{red}{q}} \\ \\ 0.36q&=15 \\ \\ \frac{\cancel{0.36}q}{\cancel{\color{red}{0.36}}}&=\frac{15}{\color{red}{0.36}} \\ \\ q&=41.7 \\ \end{align}\)

The length of q is approximately 41.7 units.


Now, it is your turn! Complete the questions in your Chapter 4, Lesson 1 Practice Makes Perfect that refer to Solving Problems Involving the Tangent Ratio with a Table.



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