Trigonometry: Solving for a Side

Trigonometry: Solving for a Side

The word ‘trigonometry’ is derived from the Greek words ‘tri’(meaning three), ‘gon’ (meaning sides) and ‘metron’ (meaning measure).
Trigonometry is the study of relationships between the sides and angles of a triangle.

Applications of trigonometry:
There are an enormous number of uses of trigonometry and trigonometric functions.

1. Early astronomers used it to find out the distances of the stars and planets from the Earth.
2. In geography to measure distances between landmarks, and in satellite navigation systems.
   
3. Even today, most of the technologically advanced methods used in Engineering and Physical Sciences are based on trigonometrical concepts.
4. The sine and cosine functions are fundamental to the theory of periodic functions such as those that describe sound and light waves.

Formulas:
The formulas can be remembered by: oh heck, another hour of algebra!
The formulas can be remembered by: oscar had a heap of apples

Basic Trigonometry Rules:

1. These formulas ONLY work in a right triangle.
2. The hypotenuse is always across from the right angle.
3. Questions usually ask for an answer to the nearest units.
4. You will need a scientific or graphing calculator.

How to set up and solve a trigonometry problem when solving for a side of the triangle:

Example 1: In right triangle ABC, hypotenuse AB=15 and angle A=35º. Find leg length, BC, to the nearest tenth.
Set Up the Drawing:

1. Draw a picture depicting the situation.
2. Be sure to place the degrees INSIDE the triangle.
3. Place a stick figure at the angle as a point of reference.
4. Thinking of yourself as the stick figure, label the opposite side (the side across from you), the hypotenuse (across from the right angle), and the adjacent side (the leftover side).
   Notice how the values on the sides of the triangle “pair up”. The h pairs with the 15, the o pairs with the x, but the a stands alone. The a has no companion term. This means that the a is NOT involved in the solution of this problem. Cross it out!
5. This problem deals with o and h which means it is using sin A.

Set Up the Formula:

1. Place the degrees in the formula for angle A.
2. Replace o and h with their companion terms.
3. Using your scientific/graphing calculator, determine the value of the left side of the equation. (On most scientific calculators, press 35 first and then press the sin key. On most graphing calculators, press the sin key first and the 35 second.)
4. Solve the equation algebraically. In this case, cross multiply and solve for x. Or just remember that if the x is on the top, you will multiply to arrive at your answer. If x is on the bottom, divide to arrive at your answer (see next example).
5. Round answer to the desired value.

Example 2: In right triangle ABC, leg length BC=20 and angle B = 41º. Find hypotenuse length BA to the nearest hundredth.
Set up the diagram and the formula in the same manner as was done in
Example 1. You should arrive at the drawing and the formula shown here.

Hint: If you are having a problem solving the equation algebraically, remember that when x is on the bottom, you must divide to arrive at your answer. The division is always “divide BY the trig value decimal”.

Hint: Be sure your answer MAKES SENSE!!! The hypotenuse is always the largest side in a right triangle. So, our answer of 26.50 makes sense – it is bigger than the leg of 20.

You really know a lot of facts about these triangles:

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