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Using Trigonometry to Determine Distances and Angles

Trigonometry involves ratios of numbers. The numbers from basic trigonometry represent sides of right triangles. There are three ratios to memorize with the term "SOH-CAH-TOA."

SOH represents S = O / H or sine (angle) = opposite / hypotenuse
CAH represents C = A / H or cosine (angle) = adjacent / hypotenuse
TOA represents T = O / A or tangent (angle) = opposite / adjacent

Opposite and adjacent vary, depending upon the reference angle chosen. For instance, in the triangle below, side A is adjacent (touching) angle M, and side B is opposite (not touching) angle M. Side A is opposite angle Q, while side B is adjacent to it.

Graphic of an hypotenuse

To solve any trigonometry problem, you should:

  1. draw a right triangle and label all known parts.
  2. determine which ratio is appropriate for known side lengths
  3. write an equation with one variable
  4. solve the equation with the help of a chart or calculator
  5. check to be sure your answer makes sense
Example 1
of a right triangle

If you wanted to solve for angle x, above, and only want to use the side lengths provided, you would determine that the tangent ratio is best for this problem, because only adjacent (10) and opposite (9) values are given. tan (x) = 9/10 or tan(x) = 0.9.
With a chart or a calculator's tan -1 button, I find that the angle that satisfies the equation is approximately 42 degrees. This makes sense, as the triangle is almost isosceles, which would reveal a 45-degree angle.

Example 2
of a right triangle

In example 2, hypotenuse and opposite sides are given, so sine will be used. sin (20) = 8 / y Using a chart or the sin button on a calculator, sin(20) is found to be 0.3420. That makes the original equation the same as 0.3420 = 8 / y or 0.3420y = 8. Therefore, y equals about 23.3918. This answer seems reasonable, as the hypotenuse must be over twice as long as the shortest leg (8). It would be 16 if the angle was 60 (30-60-90 Theorem).

Example 3: When you fly your kite, you will be able to physically measure the angle at which the kite is flying and the length of the string, but will not be able to measure its height without trigonometry.

of a triangle

Here is an example of what you may have to work with:
You are flying your kite at a 63-degree angle of elevation (relative to the horizon), and have 132 feet of kite string out. How high is your kite above he ground, if you are holding the string 3 feet off of the ground?

To solve, first draw the picture:

90 - 63 = 27, which is the angle inside the triangle, where the person is standing. This will be the angle we will use in our equation. Hypotenuse and adjacent are to be used, so we will be working with cosine. cos (27) = x/132 or 0.8910 = x/132 or x = 117.6120. This makes sense, as the height should be less than the hypotenuse and a little more than square root (3)/2 times the hypotenuse (theheight for a 30-60-90 triangle). Since the right triangle formed by the string and the height of the kite is held 3 feet off the ground, you must add that amount to the height you calculated. 117.6120 + 3 = 120.6120 feet.

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