Graphs of the Trigonometric Functions

The unit-circle definition is a snapshot. It tells you $\sin\theta$ and $\cos\theta$ for one angle at a time. But what does the function look like as $\theta$ varies? If you let the angle sweep forward and plot the height as a function of time, the unit circle unrolls into a wave. The graphs of sine, cosine, and tangent are that unrolling. Once you see the wave, the period, the maxima, the zeros, and the asymptotes all stop being formulas to memorise and start being features of a picture.

The sine wave

Set the $y$-axis equal to $\sin\theta$ and let $\theta$ run along the $x$-axis. The point sweeping the unit circle drops its $y$-coordinate onto the plot, sketching a curve that starts at $(0, 0)$, rises to $(\pi/2, 1)$, falls back to $(\pi, 0)$, plunges to $(3\pi/2, -1)$, and returns to $(2\pi, 0)$. Then it repeats. The horizontal length of one cycle is the period, and for plain $y = \sin x$ the period is $2\pi$. The maximum displacement from the axis, here $1$, is the amplitude.

The cosine wave

The same exercise with the $x$-coordinate gives $y = \cos x$: same wave shape, but shifted, it starts at $(0, 1)$, hits zero at $\pi/2$, falls to $-1$ at $\pi$, and so on. Algebraically, $\cos x = \sin(x + \pi/2)$; geometrically, cosine is sine shifted left by a quarter period.

• Music: A pure tone is $y = A \sin(2\pi f t)$; doubling the frequency $f$ raises the pitch by an octave, and amplitude $A$ is loudness. The shape of these graphs is what your ear perceives as "tone."
• Electrical Engineering: AC mains supply is a sine wave: 120 V RMS in the US is the amplitude of a 60 Hz sine; the phase shift between current and voltage in a reactive load is a real number that engineers measure directly off the graph.
• Oceanography: Tide heights through a day approximate a sine curve with period 12.4 hours; coastal-engineering safety designs read the amplitude, period, and phase shift of this graph directly from tide-gauge data.

(Press Play to watch a point sweep the unit circle on the left while its sine, cosine, or tangent value paints the graph on the right. Switch between the three modes; drag the slider to scrub.)

The tangent graph

The tangent function $\tan x = \sin x / \cos x$ has a different shape and a different period. It is not a smooth wave between $-1$ and $1$; it goes from $-\infty$ to $+\infty$ inside each interval $(-\pi/2, \pi/2)$, blows up at $x = \pi/2 + n\pi$ where $\cos x = 0$, and then repeats with period $\pi$ (not $2\pi$, we will explain why in the next section).

The general sinusoid: amplitude, period, phase, vertical shift

The four-parameter family $y = A\sin(Bx + C) + D$ stretches and slides the basic sine wave:

• Amplitude $= |A|$. Vertical stretch.
• Period $= \dfrac{2\pi}{|B|}$. A larger $|B|$ packs more cycles into the same horizontal distance.
• Phase shift $= -\dfrac{C}{B}$. Positive shift moves the wave to the left.
• Vertical shift $= D$. The centre line of the wave moves from $y = 0$ to $y = D$.

For instance, $y = 3\sin(2x - \pi)$ has amplitude $3$, period $\pi$, and a phase shift of $\pi/2$ to the right: factor the inside as $2(x - \pi/2)$ to see the shift directly. The same recipe works for cosine.

Periodicity in one equation

The defining property of a periodic function is $f(x + T) = f(x)$ for every $x$, where $T$ is the period. For sine and cosine, the smallest such $T$ is $2\pi$. For tangent, it is $\pi$. This is why these functions model anything that repeats: sound waves, alternating current, planetary cycles, tides.

Try it

• Predict first: what is $\sin\pi$? Set mode to sin, drag the slider to $\pi$, and verify the dot lies at $y = 0$.
• Predict first: where does $\tan x$ have its first positive vertical asymptote? Switch to tan mode and drag the slider to verify.
• Sketch $y = 2\sin(x + \pi/4)$ from memory, then verify by hand: amplitude $2$, period $2\pi$, shift $\pi/4$ left.

Pause: if you doubled the coefficient $B$ inside $\sin(Bx)$ from $1$ to $2$, does the wave get wider or narrower? Predict before checking on the widget. (The period halves, so the wave gets narrower.)

A trap to watch for

Two parameters easy to confuse: period shrinks as $B$ grows, but amplitude grows as $A$ grows. A common mistake is to read the period off the curve and write $T = 2\pi B$. It is the other way around: $T = 2\pi / B$. Sanity check on $B = 2$: the wave should complete a full cycle by $x = \pi$, not $x = 4\pi$. Always test the formula on a simple case before trusting it on a hard one.

What you now know

You can sketch $y = \sin x$, $y = \cos x$, and $y = \tan x$ from memory, identify the amplitude, period, phase shift, and vertical shift of any sinusoid $y = A\sin(Bx + C) + D$, and explain why tangent has half the period of sine. The next section dives deeper into tangent, why its period is $\pi$, where its asymptotes live, and how it connects to the slope of a line.

Quick check

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References

• Lang, S. (1971). Basic Mathematics. Springer. Chapter 11, §3, graphs of sine and cosine traced from the unit circle.
• Stewart, J. (2015). Calculus: Early Transcendentals (8th ed.). Cengage. Section 1.2 catalogs sinusoidal transformations.
• Strogatz, S. (2003). SYNC: The Emerging Science of Spontaneous Order. Hyperion. Accessible discussion of why periodic functions describe so many natural phenomena.