Geometry/Chapter 18

Trigonometry

Trigonometry is the branch of mathematics dealing with the measures of the sides and angles of a triangle. Using trigonometric ratios, one can determine the side and angle measures (i.e. "solve the triangle") using minimal information.

What is a right triangle?

A right triangle is any triangle where one angle measures 90º. A right triangle consists of two small sides, across from the smaller angles, and a longer side, across from the right angle. The sides of a right triangle have special names. The two smaller ones are called legs, and the longer side is called the hypotenuse (pronounced hi-POT-uh-noose).

Notice that the bottom-left angle is 90 degrees. The sides labelled "opposite" and "adjacent" are the legs, and the names are in reference to the angle "a". These terms will always be used when discussing sides of a right triangle, and will always be given in reference to some angle. Remember, the hypotenuse is always the longest side and is always across from the right angle.

Practice

Let's practice what we've learned. For the triangle below, fill in the blanks.

What is the length of the hypotenuse? _________
What is the length of the side opposte angle "a"? _________
What is the length of the side adjacent to angle "a"?_________

View answers

The correct answers are 10 feet, 8 feet, and 6 feet respectively.

More Practice

Rotating a right triangle does not alter the names of the sides. Consider the following triangle: Can you name the hypotenuse, adjacent and opposite sides?

(6 feet is incorrect (should be 5 feet), since h^2 = a^2 + b^2 = 9 + 16 = 25, the square root being 5)
Even though this right angle has been rotated, the sides are still named with respect to the angle "a". See if you can identify the sides of the triangle below. Remember that everything is from the point of view of angle "a".
What is the length of the hypotenuse? ___________feet
What is the length of the side opposite angle "a"? ___________feet
What is the length of side adjacent to angle "a"? ___________feet
(Two of the answers are incorrect. The opposite angle should be 3 feet, and the adjacent angle should be 4 feet)

View answers

The correct answers are 5 feet, 3 feet, 4 feet, respectively.

Trigonometric ratios

The three most common trigonometric ratios are sine (abbreviated "sin"), cosine (abbreviated "cos"), and tangent (abbreviated "tan"). These trigonometric ratios can be determined contingent on the information you have available. If the angle measure is given, entering sin(x), cos(x), or tan(x) in most scientific calculators (where "x" is the angle measure) will yield the sine, cosine, and tangent of the given value. The three other ratios are cosecant (abbreviated "csc"), secant (abbreviated "sec"), and cotangent (abbreviated "cot" or "ctn"), the reciprocal functions of the sine, cosine, and tangent respectively.

Soh Cah Toa

Figure 1. In this example, angle A is the reference angle, and so side a must be the opposite side and side c the adjacent side. Side b is the hypotenuse.

In a right triangle, if you are given at least two of the side measures, you can determine the sine, cosine, or tangent using a method commonly referred to by the mnenomic device Soh Cah Toa. What this refers to is:

Soh: sine = opposite / hypotenuse
Cah: cosine = adjacent / hypotenuse
Toa: tangent = opposite / adjacent

The terms opposite and adjacent are in relation to the reference angle, which is the angle your calculations are based upon. In Figure 1, since angle A is the reference angle, then the side opposite of that angle is side a. While there are two sides adjacent to angle A, side b is designated as the hypotenuse so side c is the adjacent side. Therefore, in Figure 1:

sin(A) = a/b
cos(A) = c/b
tan(A) = a/c

Please note that "sin(A)" is read as "sine of A," "cos(A)" is read as "cosine of A," and "tan(A)" is read as "tangent of A."

For example, if a = 7 and c = 14, you would need to find the tangent of angle A, as the other two ratios require the hypotenuse which is unavailable. As the tangent is the opposite divided by the adjacent, you would get tan(A) = 7/14, which can be simplified to one-half.

What now?

Note: Please make sure first that your calculator gives angle measures in degrees.

When determining the measures of a triangle, the sines, cosines, and tangents have a function: determining the angle measure. Above, it was determined that the tangent of the reference angle, A, is 1/2. If you entered tan^-1(1/2) into a scientific calculator, you will see that you will get the angle measure of A, which is approximately 26.6°. You now have two angle measures: 26.6° and 90°, since this is a right triangle. Since the sum of all angles in a triangle is 180°, subtract the two known angle measures from 180 and you will get the third angle measure: 63.4°.

Solving the side and angle measures of triangles will be covered in more depth in Chapter 19.