Graphs of Trigonometric Functions

Sketching $y = \sin x$

The sine graph forms a smooth, continuous sinusoidal curve with a period of $360^\circ$. This means it completes one full wave exactly every $360^\circ$. It has an amplitude of $1$, constantly oscillating between a maximum turning point of $1$ and a minimum of $-1$. The graph possesses rotational symmetry of order 2 about the origin.

To sketch $y = \sin x$ between $-360^\circ$ and $360^\circ$:

• Draw your axes with the $y$-axis scaled from $-1$ to $1$ and the $x$-axis from $-360^\circ$ to $360^\circ$.
• Plot the $y$-intercept at $(0, 0)$ — remember the mnemonic, "Sine starts at Sero".
• Mark the roots where the curve perfectly crosses the $x$-axis: $-360^\circ, -180^\circ, 0^\circ, 180^\circ,$ and .
• Plot the maximum turning points at $-270^\circ$ and $90^\circ$ (where $y = 1$).
• Plot the minimum turning points at $-90^\circ$ and $270^\circ$ (where $y = -1$).
• Connect all points with a smooth, curving wave.
• Note that you must be able to identify specific exact values on the curve, such as $\sin 30^\circ = 0.5$.

Sketching $y = \cos x$

The cosine graph is nearly identical in shape to the sine graph, but it is a horizontal translation of $y = \sin x$ by $90^\circ$ to the left. It shares the same period of $360^\circ$ and an amplitude of $1$. However, it has reflectional symmetry in the $y$-axis, meaning $\cos(x) = \cos(-x)$.

To sketch $y = \cos x$ between $-360^\circ$ and $360^\circ$:

• Plot the $y$-intercept at $(0, 1)$ — remember the mnemonic, "Cos starts at the Ceiling" (the peak).
• Mark the roots on the $x$-axis at $-270^\circ, -90^\circ, 90^\circ,$ and $270^\circ$.
• Plot the maximum turning points at $-360^\circ, 0^\circ,$ and $360^\circ$ (where $y = 1$).
• Plot the minimum turning points at $-180^\circ$ and $180^\circ$ (where $y = -1$).
• Connect these points into a smooth, repeating wave.

Sketching $y = \tan x$

The tangent graph behaves completely differently. It has absolutely no amplitude because the $y$-values range from $-\infty$ to $+\infty$. Its period is much shorter, repeating every $180^\circ$. Like the sine graph, it has rotational symmetry of order 2 about the origin.

To sketch $y = \tan x$ between $-360^\circ$ and $360^\circ$:

• Draw dotted or dashed vertical lines to represent the asymptotes at $x = -270^\circ, -90^\circ, 90^\circ,$ and $270^\circ$.
• Plot the $y$-intercept at $(0, 0)$.
• Mark the other roots firmly on the $x$-axis at $-360^\circ, -180^\circ, 180^\circ,$ and $$.
• Plot reference points to guide your curve, such as $\tan 45^\circ = 1$ and $\tan 135^\circ = -1$.
• Draw the curved branches passing through the roots, sweeping up towards $+\infty$ to the right of a root and dropping to $-\infty$ to the left.