Graphs of Other Trig Functions

While sine and cosine make smooth, gentle waves, the other four trig functions have a wilder personality — they shoot off toward infinity at certain points. These "blow-up" points are called vertical asymptotes, and they're invisible walls the graph can never cross. Tangent and secant blow up wherever cosine is zero; cotangent and cosecant blow up wherever sine is zero.

The biggest thing to remember is the period difference. Tangent and cotangent are "impatient" — they repeat every $\frac{\pi}{|B|}$ instead of the usual $\frac{2\pi}{|B|}$. That means their pattern is twice as fast as sine and cosine. Secant and cosecant keep the same $\frac{2\pi}{|B|}$ period as the sine/cosine waves they're built from.

Tangent rises from $-\infty$ to $+\infty$ between each pair of asymptotes — it always goes uphill. Cotangent does the opposite — it always goes downhill. Secant and cosecant form U-shaped branches that open up and down, hugging the peaks and valleys of cosine and sine respectively.

When graphing a transformed version like $y = A\tan(Bx - C) + D$, the same A-B-C-D ideas from sine and cosine apply — just swap in the correct period formula. The most important step is finding where the asymptotes land, because they determine where each branch starts and ends. If there's a phase shift, the asymptotes shift too.