15.4. Graphs of the Other Trigonometric Functions

15.4.1. The Cosine Wave

▪ Exploration:

Try this. Graph at least one cycle of y = sin x. Then on the same axes, graph y = cos x.

How are the two waves similar? How do they differ? How would you obtain one from the other?

15.4.2. Cosine and Sine Curves Related

Note in Fig. 15-19 and in the calculator screen that the cosine curve and the sine curve have the same shape. In fact, the cosine curve appears to be identical to a sine curve shifted 90° to the left, or

cos θ = sin(θ + 90°)

We can show that Eq. 1 is true. We lay out the two angles θ and θ + 90° (Fig. 15-20), choose points P and Q so that OP = OQ, and drop perpendiculars PR and QS to the x axis. Since triangles OPR and OQS are congruent, we have OR= QS. The cosine of θ is then

which verifies Eq. 1.

Calculator graphs of y = sin x, shown light, and y = cos x, shown heavy. Tick marks are 30° apart on the x axis and one unit apart on the y axis.

Figure 15.19. Manual graphs of y = sin x and y = cos x.
Figure 15.20. FIGURE 15-20

15.4.3. Graph of the General Cosine Function

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