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Finding missing side lengths using trigonometric ratios

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

A right triangle with sides a, c, and 7. Angle of 30 degrees is also labeled which is opposite the side labeled 7.

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

tan ( 30° ) = 7 a

We rearrange to solve for a .

a = 7 tan ( 30° ) 12.1

We can use the sine to find the hypotenuse.

sin ( 30° ) = 7 c

Again, we rearrange to solve for c .

c = 7 sin ( 30° ) 14
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A right triangle has one angle of π 3 and a hypotenuse of 20. Find the unknown sides and angle of the triangle.

adjacent = 10 ; opposite = 10 3 ; missing angle is π 6

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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] .

Diagram of a radio tower with line segments extending from the top and base of the tower to a point on the ground some distance away. The two lines and the tower form a right triangle. The angle near the top of the tower is the angle of depression. The angle on the ground at a distance from the tower is the angle of elevation.

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.

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 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.

A tree with angle of 57 degrees from vantage point. Vantage point is 30 feet from tree.

We know that the angle of elevation is 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 57° , letting h be the unknown height.

tan  θ = opposite adjacent tan ( 57° ) = h 30 Solve for  h . h = 30tan ( 57° ) Multiply . h 46.2 Use a calculator .

The tree is approximately 46 feet tall.

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Questions & Answers

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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