Sine Graph: y = sin x

The equation \(y = \sin x\) describes one of the most important curves in trigonometry. If you plot every value of \(\sin x\) as \(x\) runs from negative angles through positive angles, you get a smooth, repeating wave called the sine curve. Understanding this graph is the foundation for later work with amplitude, period, phase shift, and identifying trig equations from graphs.

Before graphing, it helps to already be comfortable using the sine ratio in a right triangle, since the graph is really just a picture of how that ratio changes as the angle changes.

Picture a point moving counterclockwise around the unit circle, starting at \((1, 0)\). At any angle \(x\) (measured in radians), the point's height above the x-axis is exactly \(\sin x\). As \(x\) increases, that height rises to a maximum of \(1\), falls back through \(0\) to a minimum of \(-1\), then returns to \(0\), and the whole pattern repeats. Plotting that height against the angle \(x\) produces the sine graph.

Because the circle is measured in radians, it is worth being confident converting degrees and radians before working through this section, since angles on the sine graph are almost always given in radians.

Every sine graph you sketch should show these five features:

• Domain: all real numbers, since \(\sin x\) is defined for every angle \(x\).
• Range: \(-1 \le y \le 1\), because the height on the unit circle never goes above \(1\) or below \(-1\).
• Amplitude: \(1\), the distance from the midline (\(y = 0\)) up to the maximum.
• Period: \(2\pi\), the length of \(x\)-values it takes for the wave to repeat one full cycle.
• Intercepts: the graph crosses the x-axis at \(x = 0, \pi, 2\pi, -\pi, -2\pi, \dots\), that is, every multiple of \(\pi\).

A quick way to sketch \(y = \sin x\) by hand is to plot five key points that mark the maximum, minimum, and each x-intercept over one period, then connect them with a smooth curve.

1. Draw and label the x-axis in multiples of \(\pi\) (usually \(\frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi\)) and the y-axis from \(-1\) to \(1\).
2. Plot the five key points: \((0, 0)\), \(\left(\frac{\pi}{2}, 1\right)\), \((\pi, 0)\), \(\left(\frac{3\pi}{2}, -1\right)\), and \((2\pi, 0)\).
3. Connect the points with one smooth, continuous curve, never with straight lines or sharp corners.
4. Extend the pattern left and right by repeating the same shape every \(2\pi\) units, since the sine function is periodic.

Consider \(y = 2\sin x\). Every output of \(\sin x\) is now multiplied by \(2\), so the curve still crosses zero at the same x-intercepts, but it now rises to a maximum of \(2\) and falls to a minimum of \(-2\) instead of \(1\) and \(-1\). The period stays \(2\pi\) because multiplying the whole function by a number outside of \(\sin\) changes height, not width.

Now consider \(y = \sin(2x)\). Doubling the input speeds up the cycle, so the wave now completes a full period in \(\pi\) units instead of \(2\pi\). The amplitude is unchanged, still \(1\), but the graph looks compressed horizontally compared to \(y = \sin x\).

The sine graph is often taught side by side with the cosine graph, which has the same amplitude and period but is shifted horizontally. Once you are comfortable with \(y = \sin x\), it is worth comparing it to the reciprocal graphs built from sine and cosine, such as the cosecant graph, which has vertical asymptotes wherever \(\sin x = 0\).

• Do not connect the key points with straight line segments; the sine curve is smooth and rounded at every peak and trough.
• Do not confuse period with amplitude; amplitude controls height, period controls how often the wave repeats.
• Remember the graph is defined for all real \(x\), so it never stops or has gaps, unlike graphs with asymptotes.

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