Graphs of Tangent, Cotangent, Secant, and Cosecant

The sine and cosine graphs are smooth, continuous waves. The other four trig functions behave differently — they have vertical asymptotes where they are undefined, and their shapes are distinct. Understanding these graphs completes your picture of all six trig functions and is essential for advanced work in AC circuit analysis, structural engineering, and any field that relies on periodic behavior.

The Tangent Function

The tangent function is defined as $\tan(x) = \frac{\sin(x)}{\cos(x)}$. Whenever $\cos(x) = 0$, tangent is undefined — and that is where the vertical asymptotes appear.

Key properties of $y = \tan(x)$:

• Period: $\pi$ (not $2\pi$) — the tangent repeats twice as fast as sine and cosine
• Vertical asymptotes: at $x = \frac{\pi}{2} + n\pi$ for any integer $n$ (wherever $\cos(x) = 0$)
• Passes through the origin with slope 1 — the tangent line to the curve at $(0, 0)$ has slope 1
• Range: all real numbers $(-\infty, \infty)$
• Odd function: $\tan(-x) = -\tan(x)$, so the graph is symmetric about the origin

Between each pair of asymptotes, the tangent curve rises from $-\infty$ to $+\infty$, passing through zero at every multiple of $\pi$. Each S-shaped segment is one complete period.

The Cotangent Function

The cotangent function is $\cot(x) = \frac{\cos(x)}{\sin(x)}$, the reciprocal of tangent. It is undefined wherever $\sin(x) = 0$.

Key properties of $y = \cot(x)$:

• Period: $\pi$ (same as tangent)
• Vertical asymptotes: at $x = n\pi$ for any integer $n$ (wherever $\sin(x) = 0$)
• Decreasing: between each pair of asymptotes, cotangent falls from $+\infty$ to $-\infty$ — the opposite direction of tangent
• Range: all real numbers $(-\infty, \infty)$
• Odd function: $\cot(-x) = -\cot(x)$

You can think of cotangent as a reflected and shifted tangent. Specifically:

$\cot(x) = -\tan\!\left(x - \frac{\pi}{2}\right)$

This means the cotangent graph is the tangent graph reflected across the horizontal axis and shifted right by $\frac{\pi}{2}$. While tangent passes through the origin heading upward, cotangent passes through $\left(\frac{\pi}{2}, 0\right)$ heading downward.

The key difference to remember: tangent has asymptotes at odd multiples of $\frac{\pi}{2}$, while cotangent has asymptotes at multiples of $\pi$ (including zero).

The Secant Function

The secant function is $\sec(x) = \frac{1}{\cos(x)}$. Since it is the reciprocal of cosine, it shares the same asymptote locations as tangent (wherever $\cos(x) = 0$), but its shape is entirely different — instead of S-curves, it forms U-shaped parabola-like curves opening upward and downward.

Key properties of $y = \sec(x)$:

• Period: $2\pi$ (same as cosine)
• Vertical asymptotes: at $x = \frac{\pi}{2} + n\pi$ (same as tangent)
• Range: $(-\infty, -1] \cup [1, \infty)$ — secant is never between $-1$ and $1$
• Even function: $\sec(-x) = \sec(x)$, symmetric about the $y$-axis
• Where $\cos(x) = 1$, $\sec(x) = 1$ (the curves touch)
• Where $\cos(x) = -1$, $\sec(x) = -1$ (the curves touch)

Notice how the secant curve touches the cosine curve at every peak and trough — at those points, $\cos(x) = \pm 1$ and $\sec(x) = \pm 1$. Between those contact points, the secant curves away to infinity as cosine approaches zero.

The upward-opening U-shapes appear wherever cosine is positive, and the downward-opening U-shapes appear wherever cosine is negative. The secant graph never enters the horizontal band between $y = -1$ and $y = 1$.

The Cosecant Function

The cosecant function is $\csc(x) = \frac{1}{\sin(x)}$. It relates to sine in the same way that secant relates to cosine.

Key properties of $y = \csc(x)$:

• Period: $2\pi$ (same as sine)
• Vertical asymptotes: at $x = n\pi$ (wherever $\sin(x) = 0$) — same locations as cotangent
• Range: $(-\infty, -1] \cup [1, \infty)$ — never between $-1$ and $1$, just like secant
• Odd function: $\csc(-x) = -\csc(x)$
• U-shaped curves opening up and down, same shape as secant but shifted

The relationship between cosecant and secant mirrors the relationship between sine and cosine. Since $\sin(x) = \cos\!\left(x - \frac{\pi}{2}\right)$, we have:

$\csc(x) = \sec\!\left(x - \frac{\pi}{2}\right)$

The cosecant graph is the secant graph shifted right by $\frac{\pi}{2}$. Where sine reaches its peak of $1$, cosecant has a minimum of $1$. Where sine hits its trough of $-1$, cosecant has a maximum of $-1$.

Summary Comparison

Notice the pattern: tangent and secant share the same asymptote locations (where $\cos = 0$), while cotangent and cosecant share the same asymptote locations (where $\sin = 0$).

Transformations

The same transformation rules you learned for sine and cosine graphs apply to all six trig functions. The general transformed tangent function is:

$y = A\tan(B(x - C)) + D$

• Vertical stretch $A$: multiplies all $y$-values by $A$ (if $A$ is negative, the curve flips vertically)
• Period: $\frac{\pi}{|B|}$ for tangent and cotangent, $\frac{2\pi}{|B|}$ for secant and cosecant
• Phase shift $C$: shifts the graph (and its asymptotes) left or right
• Vertical shift $D$: moves the entire graph up or down

For example, $y = 3\tan(2x)$ has period $\frac{\pi}{2}$ (asymptotes twice as close together) and a vertical stretch of 3. The function $y = \sec(x - \pi) + 1$ shifts the standard secant graph right by $\pi$ and up by 1.

When you transform secant or cosecant, remember that the forbidden band (the gap where no values exist) shifts with $D$. For $y = A\sec(Bx) + D$, the graph never takes values between $D - |A|$ and $D + |A|$.

Common Mistakes

Mixing up asymptote locations. Tangent and secant have asymptotes at $x = \frac{\pi}{2} + n\pi$ (odd multiples of $\frac{\pi}{2}$). Cotangent and cosecant have asymptotes at $x = n\pi$ (integer multiples of $\pi$). A quick check: tangent is undefined where cosine is zero, cotangent is undefined where sine is zero.

Forgetting the forbidden range. Secant and cosecant never take values between $-1$ and $1$. If you compute $\sec(\theta) = 0.5$, you have made an error — that value is impossible.

Drawing continuous curves across asymptotes. The secant and cosecant graphs are not connected across asymptotes. Each U-shaped segment is a separate piece of the graph. Similarly, each S-shaped tangent/cotangent segment is separate. Never draw a line crossing an asymptote.

Using period $2\pi$ for tangent and cotangent. The tangent and cotangent functions repeat every $\pi$, not every $2\pi$. This is one of the most common errors when setting up graphs or solving equations.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Tangent and cotangent have period $\pi$ and range all real numbers — they are the “S-curve” trig functions
• Secant and cosecant have period $2\pi$ and range $(-\infty, -1] \cup [1, \infty)$ — they are the “U-curve” trig functions
• Vertical asymptotes occur where the denominator function ($\cos$ or $\sin$) equals zero
• Tangent/secant share asymptote locations; cotangent/cosecant share asymptote locations
• The same transformation rules (amplitude, period, phase shift, vertical shift) apply to all six trig functions
• These functions are never drawn as continuous curves across their asymptotes — each segment is a separate branch

Return to Trigonometry for more topics in this section.