MTH5P1: Trigonometric Ratios in a Triangle

Trigonometric Ratios in a Triangle

Definition of sin a
Definition: For any acute angle a, we draw a right triangle that includes a.
The sine of a, abbreviated sin a, is the ratio of the length of the leg opposite this angle to the length of the hypotenuse of the triangle.

For example, in the right triangle ABC (diagram above), sin a= a/c.
We can see immediately that this definition has a weak point: it does not tell us exactly which right triangle to draw.

There are many right triangles, large ones and small ones that include a given angle a.
Let us try to answer the following questions.

Example 12 Find sin 30°.
“Solution” 1. Formally, we are not obliged to solve the problem, since we are given only the measure of the angle, without a right triangle that includes it.

We know, from geometry, that whatever the value of the hypotenuse, the side opposite the 30° angle will be half this value, I so sin 30° will always be 1/2. This value depends only on the measure of the angle, and not on the lengths of the sides of the particular triangle we used.

The French student measured E F with his ruler, then measured ED, then took the ratio E F jED and sent the answer to his American friend. A
I A theorem in geometry tells us that in a right triangle with a 30° angle the side opposite this angle is half the hypotenuse.

Definition of sin a.
few days later, he woke up in the middle of the night and realized, “Sacre bleu! I forgot that Americans use inches to measure lengths, while we use centimeters. I will have to tell my friend that I gave her the wrong answer!”
What must the French student do to correct his answer?
Solution. He does not have to do anything – the answer is correct. The sine of an angle is a ratio of two lengths, which does not depend on any unit of measurement.

For example, if one segment is double another when measured in centimeters, it is also double the other when measured in inches.
In general, for any angle of ao (for 0 < a < 90), the value of sin a depends only on a, and not on the right triangle containing the angle.

This is true because any two triangles containing acute angle a are similar, so the ratios of corresponding sides are equal. Sin a is merely a name for one of these ratios.

Example 12 shows that the value of sino: does not depend on the particular triangle which contains a. Example 13 shows that the value of sin a does not depend on the unit of measurement for the sides of the triangle. In fact, we can examine Example 12 more closely.

To determine the value of sin 30°, we need three pieces of information: (a) the angle; (b) the right triangle containing the angle; (c) the unit of measurement for the sides of the triangle. We have just shown that the value of sin a does not in fact depend on the last two pieces of information.

In the following list, cross off each number that is less than the sine of 60°. Then check your work .with a calculator.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Hint: Remember the relationships among the sides of a 30-60-90 triangle.

Find the hidden sine
Sometimes the sine of an angle lurks in a diagram where it is not easy to spot. The following exercises provide practice in finding ratios equal to the sine of an angle, and lead to some interesting formulas.

The diagram below shows a right triangle with an altitude drawn to the hypotenuse. The small letters stand for the lengths of certain line segments.

a)  Find a ratio of the lengths of two segments equal to sin a.
b) Find another ratio of the lengths of two segments equal to sin a.
c) Find a third ratio of the lengths of two segments equal to sin a.
2. The three angles of triangle ABC below are acute (in particular, none
of them is a right angle), and CD is the altitude to side A B. We let
CD= h, and CA =b.

The cosine ratio:
Definition: In a right triangle with the acute angle a, the ratio of the leg adjacent to angle a to the hypotenuse is called the cosine of angle a, abbreviated cos a.

Notice that the value of cos a, like that of sin a, depends only on a and not on the right triangle that includes a. Any two such triangles will be similar, and the ratio cos a will thus be the same in each.

The diagram below shows a right triangle with an altitude drawn to the hypotenuse. The small letters stand for the lengths of certain line segments.

a) Find a ratio of the lengths of two segments equal to cos a.
b) Find another ratio of the lengths of two segments equal to cos a.
c) Find a third ratio of the lengths of two segments equal to cos a.

relation between the sine and the cosine

1. Show that sin 29° = cos 61 o.
2. If sin 35° = cos x, what could the numerical value of x be?
3. Show that we can rewrite the theorem of the above section as: sin a =cos (90- a).

If you look carefully among the exercises of the previous section, you will see examples of the following result:

Our next best friends (and the sine ratio)
It is usually not very easy to find the sine of an angle, given its measure.
But for some special angles, it is not so difficult. We have already seen that sin 30° = 1/2.
Example 15 Find cos 30°.
Solution. To use our definition of the cosine of an angle, we must draw a right triangle with a 30° angle, a triangle with which we are already friendly.

Example 16 Show that cos 60° = sin 30°.
Solution. In the 30-60-90 triangle we’ve drawn above, one acute angle is 30°, and the other is 60°. Standing on the vertex of the 30° angle, we see that the opposite leg has length 1, and the hypotenuse has length 2.
Thus sin 30° = 1/2. But if we walk over to the vertex of the 60° angle, the opposite leg becomes the adjacent leg, and we see that the ratio that was sin 30° earlier is also cos 60°.

Fill in the following table. You may want to use the model triangles given in the diagram below.

What is the value of sin 90°?
So far we have no answer to this question: We defined sin a only for an acute angle. But there is a reasonable way to define sin 90°.

The picture below shows a series of triangles with the same hypotenuse, but with different acute angles a:

As the angle a gets larger, the ratio of the opposite side to the hypotenuse approaches 1. So we make the following definition.
Definition sin 90° = 1.
The diagram above also suggests something else about sin a. Remember that the hypotenuse of a right triangle is longer than either leg. Since sin a is the ratio of a leg of a right triangle to its hypotenuse, sin a can
never be larger than 1. So if someone tells you that, for a certain angle a, sin a = 1.2 or even 1.01, you can immediately tell him or her that a mistake has been made.

An exploration: How large can the sum be ?
The same series of triangles lets us make a definition for cos 90°.

As the angle a gets closer and closer to 90°, the hypotenuse remains the same length, but the adjacent leg gets shorter and shorter. This same diagram leads us to the following definition.

As before, these ratios depend only on the size of the angle a, and not on the lengths of the sides of the particular triangle we are using, or on how we measure the sides.

The following theorem generalizes our statement of this fact for sin a.
Theorem The values of the trigonometric ratios of an acute angle depend only on the size of the angle itself, and not on the particular right triangle containing the angle.
Proof Any two triangles containing a given acute angle are similar, so ratios of corresponding sides are equal.

The trigonometric ratios are just names for these ratios.