5.4 Right triangle trigonometry  (Page 5/12)

Measuring a distance indirectly

To find the height of a tree, a person walks to a point 30 feet from the base of the tree. She measures an angle of $\mathrm{57°}$ between a line of sight to the top of the tree and the ground, as shown in [link] . Find the height of the tree.

We know that the angle of elevation is $\mathrm{57°}$ and the adjacent side is 30 ft long. The opposite side is the unknown height.

The trigonometric function relating the side opposite to an angle and the side adjacent to the angle is the tangent. So we will state our information in terms of the tangent of $\mathrm{57°},$ letting $h$ be the unknown height.

$$\begin{array}{ll}\tan \theta =\frac{\text{opposite}}{\text{adjacent}}\hfill & \hfill \\ \text{tan}(\mathrm{57°})=\frac{h}{30}\hfill & \text{Solve for }h.\hfill \\ h=30\tan (\mathrm{57°})\hfill & \text{Multiply}.\hfill \\ h\approx 46.2\hfill & \text{Use a calculator}.\hfill \end{array}$$

The tree is approximately 46 feet tall.

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Key equations

Key concepts

• We can define trigonometric functions as ratios of the side lengths of a right triangle. See [link] .
• The same side lengths can be used to evaluate the trigonometric functions of either acute angle in a right triangle. See [link] .
• We can evaluate the trigonometric functions of special angles, knowing the side lengths of the triangles in which they occur. See [link] .
• Any two complementary angles could be the two acute angles of a right triangle.
• If two angles are complementary, the cofunction identities state that the sine of one equals the cosine of the other and vice versa. See [link] .
• We can use trigonometric functions of an angle to find unknown side lengths.
• Select the trigonometric function representing the ratio of the unknown side to the known side. See [link] .
• Right-triangle trigonometry permits the measurement of inaccessible heights and distances.
• The unknown height or distance can be found by creating a right triangle in which the unknown height or distance is one of the sides, and another side and angle are known. See [link] .

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