AP® Precalculus · Unit 3 · Topic 3.4
The Cosine Graph
The cosine graph tracks the horizontal position of a point circling the unit circle, so it starts at its maximum. It is the sine graph slid left by π/2.
Picture a point moving counterclockwise around the unit circle starting at (1, 0). Sine records how high that point is off the x-axis, so at theta = 0 it starts at zero and climbs. Cosine records how far the point is from the y-axis, left to right. At theta = 0 the point sits at its farthest right position, so cosine starts at its maximum value of 1, not zero. That single difference, height versus horizontal distance, is the whole reason the two graphs look shifted from each other.
As theta sweeps from 0 to 2pi, the horizontal displacement x = cos(theta) starts at 1, eases down through 0 at theta = π/2, hits -1 at theta = π, climbs back through 0 at 3pi/2, and returns to 1 at 2pi. Plot those values against theta and you get a wave with the same shape as sine, same height, same period 2pi, just caught at a different point in its cycle. That shape match is not a coincidence: cos(theta) = sin(theta + π/2) says that cosine's value at angle theta equals sine's value a little further along, at theta + π/2. On a graph that means the cosine curve is the sine curve slid π/2 units to the left.
This slide connects directly to the transformation work surrounding both graphs: once you can slide, stretch, and reflect sine, the same tools describe cosine, and any phase shift you meet later is just this same idea with a different amount of slide.
Key terms
- cosine
- The horizontal displacement, or x-coordinate, of a point at angle theta on the unit circle.
- phase shift
- A horizontal slide of a trigonometric graph, here π/2 units, that turns sine into cosine.
- period
- The length of one full repeating cycle of a wave, equal to 2pi for both sine and cosine.
- unit circle
- A circle of radius 1 centered at the origin, used to define sine and cosine for any angle.
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End of topic.