This chapter covers the six trigonometric functions and solving right triangles.

EO 1.2Given the following trigonometric terms, IDENTIFY the related function:

a. Sine b. Cosine c. Tangent d. Cotangent e. Secant f. Cosecant

EO 1.3Given a problem, APPLY the trigonometric functions to solve for the unknown.

As shown in the previous chapter, the lengths of the sides of right triangles can be solved using the Pythagorean theorem. We learned that if the lengths of two sides are known, the length of the third side can then be determined using the Pythagorean theorem. One fact about triangles is that the sum of the three angles equals 180. If right triangles have one 90 angle, then the sum of the other two angles must equal 90. Understanding this, we can solve for the unknown angles if we know the length of two sides of a right triangle. This can be done by using the six trigonometric functions.

In right triangles, the two sides (other than the hypotenuse) are referred to as the opposite and adjacent sides. In Figure 2, side a is the opposite side of the angle 0 and side b is the adjacent side of the angle . The terms hypotenuse, opposite side, and adjacent side are used to distinguish the relationship between an acute angle of a right triangle and its sides. This relationship is given by the six trigonometric functions listed below:

Figure 2 Right Triangle

The trigonometric value for any angle can be determined easily with the aid of a calculator. To find the sine, cosine, or tangent of any angle, enter the value of the angle into the calculator and press the desired function. Note that the secant, cosecant, and cotangent are the mathematical inverse of the sine, cosine and tangent, respectively. Therefore, to determine the cotangent, secant, or cosecant, first press the SIN, COS, or TAN key, then press the INV key.

Example:

Determine the values of the six trigonometric functions of an angle formed by the x-axis and a line connecting the origin and the point (3,4).

Solution:

To help to "see" the solution of the problem it helps to plot the points and construct the right triangle.

Label all the known angles and sides, as shown in Figure 3.

From the triangle, we can see that two of the sides are known. But to answer the problem, all three sides must be determined. Therefore the Pythagorean theorem must be applied to solve for the unknown side of the triangle.

Figure 3 Example Problem

Having solved for all three sides of the triangle, the trigonometric functions can now be determined. Substitute the values for x, y, and r into the trigonometric functions and solve.

Although the trigonometric functions of angles are defined in terms of lengths of the sides of right triangles, they are really functions of the angles only. The numerical values of the trigonometric functions of any angle depend on the size of the angle and not on the length of the sides of the angle. Thus, the sine of a 30 angle is always 1/2 or 0.500.