Technical Mathematics, Sixth Edition

7.1. The Trigonometric Functions

We introduced the right triangle in our chapter on geometry and continue with it here. The difference is that now we will work with the angles of a triangle, and not just the sides, as we did before.

7.1.1. Sine, Cosine, and Tangent

▪ Exploration:

Try this. Draw a right triangle with sides of any length. Do this with drafting instruments or with a CAD program. Then

• Find the ratio of any two sides by measuring them and dividing one by the other. Recall that a ratio of two quantities is their quotient, that is, one divided by the other.

• Next, enlarge the triangle on a photocopier and find the ratio of the same two sides.

• Then, reduce the triangle on a photocopier and find the ratio of the same two sides.

• Rotate, move, or flip your triangle, without changing the magnitude of the angles, and again find the ratio of the same two sides.

What do you conclude? Repeat by taking the ratio of two other sides. Are your conclusions the same?

You may have found that if you do not change the angles of a right triangle, the ratios of the sides are always the same, regardless of how long the sides are.

For the right triangle of Fig. 7-2, we first note that one side is opposite to acute angle θ and the other side is adjacent to θ, and we recall that the hypotenuse is always the side opposite the right angle. There are six ways to form ratios of the three sides. The three most important are defined as follows:

• The sine of angle θ (sin θ) is the ratio of the opposite ...