Rotation
The trigonometric ratios all translate into functions that can generate distinct patterns when applied to a Cartesian plane. As you increase the values you introduce to the functions, the patterns change in restricted, predictable ways. The most ready way to generate such patterns involves translating the three sides of the standard triangle (opposite, adjacent, and hypotenuse) so that you can understand them in relation to the coordinates you plot on the Cartesian plane (x, y). Work earlier in this chapter anticipates this activity. Table 12.4 shows you how the sides of a triangle relate to the values you generated using values typical of your work with the Cartesian plane. In each instance, you work with a standard triangle.
| Item | Ratio | Mnemonic |
|---|---|---|
| Sine |  |  |
| Cosine |  |  |
| Tangent |  |  |
| Cotangent |  |  |
| Secant |  |  |
| Cosecant |  |  |
As Figure 12.12 illustrates, if you use combinations of negative and positive values for x and y, you rotate the triangle around the origin of the plane through all four quadrants. The values x and y correspond to the values of x and y on the axes of the plane if the vertex of angle θ resides at (0,0).
Figure 12.12. As you rotate a standard triangle around a circle centered on the origin of the Cartesian plane, the values that describe the perimeter of the circle change from positive to negative depending on the quadrant.
