Determining Non-Permissible Values for Trig Expressions

A non-permissible value is an input value that makes a trig expression undefined. Just like you can never divide by zero in ordinary algebra, you can never let the denominator of a trig expression equal zero. Whenever that happens, the angle that caused it is excluded from the domain of the expression, so it is called "non-permissible."

This idea comes up constantly once you start working with \(\tan\theta\), \(\cot\theta\), \(\sec\theta\), \(\csc\theta\), and any rational expression built from sine and cosine. Before you can confidently solve equations or simplify identities involving these functions, you need to know which angles are off-limits.

All four of these functions are actually built as fractions using sine and cosine:

\(\tan\theta = \dfrac{\sin\theta}{\cos\theta}\), \(\quad \cot\theta = \dfrac{\cos\theta}{\sin\theta}\), \(\quad \sec\theta = \dfrac{1}{\cos\theta}\), \(\quad \csc\theta = \dfrac{1}{\sin\theta}\)

Since \(\cos\theta\) appears in the denominator of \(\tan\theta\) and \(\sec\theta\), those two functions are undefined whenever \(\cos\theta = 0\). Since \(\sin\theta\) appears in the denominator of \(\cot\theta\) and \(\csc\theta\), those two are undefined whenever \(\sin\theta = 0\).

Using the exact values of trigonometric ratios for common angles, you know \(\cos\theta = 0\) at \(\theta = 90^\circ\) and \(\theta = 270^\circ\) within one full rotation, which is \(\dfrac{\pi}{2}\) and \(\dfrac{3\pi}{2}\) in radians. Because these values repeat every \(180^\circ\) (or \(\pi\) radians), the full non-permissible set for tangent and secant is written as:

\(\theta \ne 90^\circ + 180^\circ n\), or in radians, \(\theta \ne \dfrac{\pi}{2} + \pi n\), where \(n\) is any integer

Similarly, \(\sin\theta = 0\) at \(\theta = 0^\circ\) and \(\theta = 180^\circ\), repeating every \(180^\circ\), so cotangent and cosecant are undefined at:

\(\theta \ne 180^\circ n\), or \(\theta \ne \pi n\), where \(n\) is any integer

Whenever you need to find non-permissible values for any trig expression, follow the same three steps:

1. Rewrite the expression so the denominator is clearly visible, using sine and cosine if needed.
2. Set the denominator equal to zero and solve that equation for \(\theta\).
3. Write every solution as a general pattern, since the pattern repeats every \(360^\circ\) (\(2\pi\)) or every \(180^\circ\) (\(\pi\)), depending on the function.

Step two usually means going back to skills like the ASTC rule and the reference angle to figure out every angle where sine or cosine hits a particular value, and it uses the same techniques as solving first degree trigonometric equations.

Example 1: Find the non-permissible values for \(\cot\theta\).

Since \(\cot\theta = \dfrac{\cos\theta}{\sin\theta}\), the denominator is \(\sin\theta\). Setting \(\sin\theta = 0\) gives \(\theta = 0^\circ, 180^\circ, 360^\circ, \dots\), so the non-permissible values are \(\theta \ne 180^\circ n\) (or \(\theta \ne \pi n\) in radians), for any integer \(n\).

Example 2: Find the non-permissible values for \(\dfrac{1}{\sin x - 1}\).

Set the denominator equal to zero: \(\sin x - 1 = 0\), so \(\sin x = 1\). This only happens at \(x = 90^\circ\) within one rotation, and it repeats every full \(360^\circ\), so the non-permissible values are \(x \ne 90^\circ + 360^\circ n\), or \(x \ne \dfrac{\pi}{2} + 2\pi n\).

Example 3: Find the non-permissible values for \(\dfrac{\cos x}{2\sin x - 1}\).

Set the denominator to zero: \(2\sin x - 1 = 0\), so \(\sin x = \dfrac{1}{2}\). Using the reference angle of \(30^\circ\) together with the ASTC rule, sine is positive in quadrants I and II, giving \(x = 30^\circ\) and \(x = 150^\circ\) within one rotation. Since these repeat every \(360^\circ\), the non-permissible values are \(x \ne 30^\circ + 360^\circ n\) and \(x \ne 150^\circ + 360^\circ n\).

Not every non-permissible value problem involves the four reciprocal or quotient functions directly. Sometimes you are given a general rational expression, such as \(\dfrac{3}{\cos x + 1}\) or \(\dfrac{\sin x}{2\cos x - \sqrt{3}}\). The method never changes: identify the denominator, set it equal to zero, and solve. Whatever solutions come out of that equation are the values you must exclude from the domain, and every other real angle remains permissible.

These same denominator restrictions matter later when you move on to solving second degree trigonometric equations or solving double angle trig equations, since a solution that happens to fall on a non-permissible value must be thrown out even if it satisfies the original equation algebraically.

Graphing \(y = \tan x\) makes the idea of non-permissible values visual. The curve breaks apart into separate branches, and at each break the graph shoots up or down toward a vertical dashed line called an asymptote. Those vertical lines sit exactly at the non-permissible values, \(x = \dfrac{\pi}{2} + \pi n\), where \(\cos x = 0\).

Notice that between each pair of asymptotes the function behaves normally and passes through predictable points, but it can never actually touch or cross those vertical lines, because \(\tan x\) simply has no value there.

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