Sine and Cosine Graphs from Key Points

From the Unit Circle to a Graph

The unit circle presents sine and cosine as coordinates that change as an angle increases. A graph displays that same change in a different way: the angle is the horizontal coordinate, and the corresponding sine or cosine value is the vertical coordinate.

For a first graph, one full cycle runs from \(0\) to \(2\pi\). The unit-circle values at the beginning, quarter-turns, and end of this interval give five key points. These points reveal where each graph crosses the horizontal axis, where it reaches its highest and lowest values, and how its shape repeats.

Definitions and Key Angles

For each real angle \(x\), the point on the unit circle at angle \(x\) has coordinates \((\cos x,\sin x)\). Thus, in a graph of \(y=\sin x\), the angle \(x\) is the horizontal coordinate and the point’s vertical coordinate \(\sin x\) is the graph’s height. For \(y=\cos x\), the graph’s height is the point’s horizontal coordinate, \(\cos x\).

The five angles used to mark one complete turn are \(0\), \(\frac{\pi}{2}\), \(\pi\), \(\frac{3\pi}{2}\), and \(2\pi\). Their unit-circle coordinates are read in the order \((\cos x,\sin x)\). Listing both function values at these angles gives the key graph points:

The sine graph therefore starts at zero, rises to \(1\), returns to zero, falls to \(-1\), and returns to zero at the end of the cycle. The cosine graph starts at \(1\), falls to zero, reaches \(-1\), rises to zero, and returns to \(1\). Both functions stay between \(-1\) and \(1\), because each is a coordinate of a point on the unit circle.

Constructing One Cycle

A reliable sketch begins with the key points, not with a guess about the curve. Mark the horizontal positions \(0,\frac{\pi}{2},\pi,\frac{3\pi}{2},2\pi\), and mark the vertical levels \(-1,0,1\). Plot the function values at those positions. Then draw a smooth curve through the points in the order shown.

For \(y=\sin x\), the highest point is \((\frac{\pi}{2},1)\), and the lowest is \((\frac{3\pi}{2},-1)\). Its horizontal-axis intercepts over the displayed cycle are \((0,0)\), \((\pi,0)\), and \((2\pi,0)\). For \(y=\cos x\), the highest points at the cycle’s endpoints are \((0,1)\) and \((2\pi,1)\); the lowest point is \((\pi,-1)\). Its intercepts are \((\frac{\pi}{2},0)\) and \((\frac{3\pi}{2},0)\).

The endpoints \(0\) and \(2\pi\) represent the same position on the unit circle. They are both included when displaying the closed interval \([0,2\pi]\), so the graph’s value at the end of one cycle matches its value at the start. The earlier material on periodicity establishes that both standard functions repeat after \(2\pi\). Consequently, the one-cycle picture can be extended left or right without changing its shape.

Why These Points Give the Graph

The key points are not arbitrary choices. As \(x\) increases from \(0\) to \(2\pi\), the point on the unit circle makes one full revolution. Its vertical coordinate follows the sine values, and its horizontal coordinate follows the cosine values. The special angles divide that revolution into four equal quarter-turns, giving the familiar sequence of rises, falls, and intercepts.

The table fixes the function values at the selected angles, while the unit-circle motion explains how those values fit together. Between the marked angles, the graph follows the corresponding changing coordinate continuously. For a basic sketch, draw the familiar smooth wave through the exact key points; do not replace the curve with straight segments. The unit-circle picture also confirms that no value of either function rises above \(1\) or drops below \(-1\).

Worked Examples

A Common Mistake

A frequent error is to give the sine graph the cosine graph’s starting height. For example, a mistaken first step might be:

This is incorrect: the value at zero is \(\sin 0=0\), while \(\cos 0=1\). The correction is to return to the coordinate definitions: sine is the vertical coordinate of the unit-circle point \((1,0)\) at angle zero, so the sine graph starts at \((0,0)\). Cosine is the horizontal coordinate, so the cosine graph starts at \((0,1)\). Checking the first key point before sketching prevents the two graphs from being confused.

Using the Graphs

These one-cycle sketches provide a visual reference for evaluating sine and cosine, checking signs, and recognizing repeated values. A graph can also make it easy to see the locations of zeros and extrema within a cycle. When extending a graph, shift every key point by a whole number of periods: the \(y\)-value stays the same, while the \(x\)-coordinate changes by \(2\pi\) or an integer multiple of \(2\pi\).

For a careful sketch, keep the angle scale marked in radians, include the five quarter-turn positions, label the vertical levels \(-1,0,1\), and ensure the curve passes through every plotted point. The sine and cosine curves have the same range and period, but their starting values and key points differ. Those distinctions are more dependable than trying to recall the shape from memory alone.