Trigonometric Ratios
You measure angles using degrees and radians. You usually measure the lengths of sides using real numbers. You relate the measurements of angles to the measurements of sides using ratios. A set of six such ratios constitute the primary trigonometric ratios. You have already examined the ratio that generates the sine of the angle (theta) θ.
If you extend the discussion that began with the sine of the angle θ, you can then work forward to explore the ratios defined for the cosine and tangent of θ. You then move on from there to explore the ratios defined for cotangent, secant, and cosecant. Figure 12.8 illustrates the standard form of the right triangle with the sides explicitly identified.
Figure 12.8. The opposite and adjacent sides and the hypotenuse of the right triangle generate the trigonometric ratios.
Each of the trigonometric ratios provides information on the angle θ. In this respect, then, you refer to “the sine of theta,” “the cosine of theta,” and so on, and in each instance, the ratio that you explore involves a relation between two of the three sides of a right triangle. Table 12.2 details the ratios.
| Item | Ratio | Mnemonic |
|---|---|---|
| Sine |  |  |
| Cosine |  |  |
| Tangent |  |  |
| Cotangent |  |  |
| Secant |  |  |
| Cosecant |  |  |
Note
Figure 12.9 features a mnemonic diagram that might prove useful as you memorize the trigonometric ratios. The letter O designates the opposite side, the letter A designates the adjacent side, and the letter H designates the hypotenuse. Start with STS to remember sine, tangent, and secant. For the names of the other ratios, you prefix “co.” Then proceed from the notion that the sine of theta is the ratio of O/H.
Figure 12.9. Recall STS, begin with sine and O/H.
Exercise Set 12.3Show all six trigonometric ratios for right triangles with the designated values. Where the value of the hypotenuse is not given, calculate it and proceed from there.
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