Geometry: The Cosine Ratio

The Cosine Ratio

The story so far: Given one of the acute angles in a right triangle, you have studied two ratios involving the lengths of the sides of the triangle. The tangent ratio involved the length of the side opposite the angle divided by the length of the side adjacent to the angle. The sine ratio involved the length of the side opposite the angle divided by the length of the hypotenuse of the triangle. You have been playing favorites with the opposite side of the angle at the expense of the adjacent side. To even things up, let me introduce a new ratio, the cosine ratio. The cosine of an angle is the ratio of the length of the side adjacent to the angle divided by the length of the hypotenuse of the triangle. The cosine of ∠A will be abbreviated as cos ∠A.

You can play the same kinds of games that you played with the tangent and sine ratios.

• Example 4: If a right triangle has an angle with tangent ratio ⁹/₁₄ , find the sineratio and the cosine ratio of the angle.

• Solution: Because a picture is worth a thousand words when working out these problems, I have sketched this situation in Figure 20.7. Because you need to know the sine and cosine ratios of the angle, you will need to calculate the length of the hypotenuse of the triangle. You can whip out the Pythagorean Theorem and take care of that right now:
• a² + b² = c²
• 9² + 14² = c²
• c² = 277
• Now that you know the length of all three sides, finding the sine and cosine ratios can be done using the definition sin ∠A = ⁹/_√277 and cos ∠A = ¹⁴/_√277.

Figure 20.7A right triangle with tangent ratio ⁹/₁₄.

At this point, you might be tempted to rationalize the denominator and write your answers as

• sin ∠A = ^9√277/₂₇₇ and cos ∠A = ^14√277/₂₇₇.

If so, then I would be impressed with your willingness to take things one step further, fearlessly going deeper into the torrential algebraic waters to write your answer in the form that I'm sure your algebra teacher emphasized when you first learned about radicals.

Of course, it is possible that the thought of rationalizing the denominator (as the process is officially called) never even occurred to you. That's okay, too. This isn't an algebra book, and there are advantages to leaving things in such a technically improper form. (Though when I wear my algebraic hat you'll never hear—or read—me say—or write—that it's okay to leave things improperly.)

One advantage to leaving things improperly is that you can clearly see that the sine and cosine ratios are less than 1. That's a quick and easy check to see if your answers make sense. The sine and cosine ratios can be equal to 1 on special occasions, but the ratios will never be greater than 1. Remember that the tangent ratio has no such restriction. You already saw that the tangent ratio can be greater, less than, or equal to 1.

You can work out these calculations in a variety of directions. If you are given the sine, cosine, or tangent ratio of an angle, you can find the other two ratios after using the Pythagorean Theorem.