# 5.2 Right triangle trigonometry  (Page 4/12)

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## Using trigonometric functions

In previous examples, 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.

Given a right triangle, the length of one side, and the measure of one acute angle, find the remaining sides.

1. For each side, select the trigonometric function that has the unknown side as either the numerator or the denominator. The known side will in turn be the denominator or the numerator.
2. Write an equation setting the function value of the known angle equal to the ratio of the corresponding sides.
3. Using the value of the trigonometric function and the known side length, solve for the missing side length.

## Finding missing side lengths using trigonometric ratios

Find the unknown sides of the triangle in [link] .

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

$\mathrm{tan}\left(30°\right)=\frac{7}{a}$

We rearrange to solve for $\text{\hspace{0.17em}}a.$

$\begin{array}{l}a=\frac{7}{\mathrm{tan}\left(30°\right)}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=12.1\hfill \end{array}$

We can use the sine to find the hypotenuse.

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

Again, we rearrange to solve for $\text{\hspace{0.17em}}c.$

$\begin{array}{l}c=\frac{7}{\mathrm{sin}\left(30°\right)}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=14\hfill \end{array}$

A right triangle has one angle of $\text{\hspace{0.17em}}\frac{\pi }{3}\text{\hspace{0.17em}}$ and a hypotenuse of 20. Find the unknown sides and angle of the triangle.

$\text{adjacent}=10;\text{\hspace{0.17em}}$ $\text{opposite}=10\sqrt{3}\text{\hspace{0.17em}}$ ; missing angle is $\text{\hspace{0.17em}}\frac{\pi }{6}$

## 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 [link] .

Given a tall object, measure its height indirectly.

1. Make a sketch of the problem situation to keep track of known and unknown information.
2. Lay out a measured distance from the base of the object to a point where the top of the object is clearly visible.
3. At the other end of the measured distance, look up to the top of the object. Measure the angle the line of sight makes with the horizontal.
4. Write an equation relating the unknown height, the measured distance, and the tangent of the angle of the line of sight.
5. Solve the equation for the unknown height.

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