7.2.2 – Using Trigonometric Functions

In previous example, we evaluated the sine and cosine in triangles where we knew all three sides. But the real power of right-triangle trigonometry emerges when we look at triangles in which we know an angle but do not know all the sides.

Example 2 – Finding Missing Side Lengths Using Trigonometric Ratios

Find the unknown sides of the triangle in (Figure).

We know the angle and the opposite side, so we can use the tangent to find the adjacent side.

[latex]\mathrm{tan}\left(30°\right)=\frac{7}{a}[/latex]

We rearrange to solve for [latex]\,a.[/latex]

[latex]$$\begin{array}{ccc}\hfill a& =& \frac{7}{\mathrm{tan}\left(30°\right)}\hfill \\ & \approx & 12.1\hfill \end{array}$$[/latex]

We can use the sine to find the hypotenuse.

[latex]$$\mathrm{sin}\left(30°\right)=\frac{7}{c}$$[/latex]

Again, we rearrange to solve for [latex]\,c.[/latex]

[latex]$$\begin{array}{ccc}\hfill c& =& \frac{7}{\mathrm{sin}\left(30°\right)}\hfill \\ & =& 14\hfill \end{array}$$[/latex]

7.2.3 – Using Right Triangle Trigonometry to Solve Applied Problems

Right-triangle trigonometry has many practical applications. For example, the ability to compute the lengths of sides of a triangle makes it possible to find the height of a tall object without climbing to the top or having to extend a tape measure along its height. We do so by measuring a distance from the base of the object to a point on the ground some distance away, where we can look up to the top of the tall object at an angle. The angle of elevation of an object above an observer relative to the observer is the angle between the horizontal and the line from the object to the observer’s eye. The right triangle this position creates has sides that represent the unknown height, the measured distance from the base, and the angled line of sight from the ground to the top of the object. Knowing the measured distance to the base of the object and the angle of the line of sight, we can use trigonometric functions to calculate the unknown height.

Similarly, we can form a triangle from the top of a tall object by looking downward. The angle of depression of an object below an observer relative to the observer is the angle between the horizontal and the line from the object to the observer’s eye. See (Figure).

Example 3 – 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 [latex]\,57°\,[/latex] between a line of sight to the top of the tree and the ground, as shown in (Figure). Find the height of the tree.

We know that the angle of elevation is [latex]\,57°\,[/latex] 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 [latex]\,57°,[/latex] letting [latex]\,h\,[/latex] be the unknown height.

[latex]$$\begin{array}{cccc}\hfill \text{tan }\theta & =& \frac{\text{opposite}}{\text{adjacent}}\hfill & \\ \hfill \text{tan}\left(57°\right)& =& \frac{h}{30}\hfill & \phantom{\rule{1em}{0ex}}\text{Solve for }h.\hfill \\ \hfill h& =& \text{30tan}\left(\text{57°}\right)\hfill & \phantom{\rule{1em}{0ex}}\text{Multiply}\text{.}\hfill \\ \hfill h& \approx & 46.2\hfill & \phantom{\rule{1em}{0ex}}\text{Use a calculator}\text{.}\hfill \end{array}$$[/latex]

The tree is approximately 46 feet tall.