Sec 90 Degrees: Why It Is Undefined

What Is The Value Of Sec 90 Degrees?

Sec 90 degrees is undefined. There is no real number equal to $\sec 90^\circ$, because secant is defined as the reciprocal of cosine and cosine is zero at this angle:

$$\sec 90^\circ = \frac{1}{\cos 90^\circ} = \frac{1}{0} = \text{undefined}$$

The angle written in radians is $\frac{\pi}{2}$, so the same statement reads $\sec\frac{\pi}{2} = \text{undefined}$. Degrees and radians are two names for one angle, and the answer does not change between them.

One warning before anything else. Undefined is not the same as infinity. A calculator or a textbook that prints $\sec 90^\circ = \infty$ is taking a shortcut that hides what really happens near the angle, and we untangle that below in the section on the graph.

How Do You Find Sec 90 Degrees?

The fastest route uses the reciprocal identities. Secant is built directly from cosine:

$$\sec\theta = \frac{1}{\cos\theta}$$

So finding $\sec 90^\circ$ is a two-step job: find $\cos 90^\circ$ first, then take its reciprocal.

• Step 1. Read the cosine. From the trigonometric table, $\cos 90^\circ = 0$.

• Step 2. Take the reciprocal. $\sec 90^\circ = \frac{1}{0}$, and division by zero has no answer, so the value is undefined.

There is a second route through the co-function relation, and it lands in the same place. Since $\sec\theta = \csc(90^\circ - \theta)$, we get $\sec 90^\circ = \csc 0^\circ = \frac{1}{\sin 0^\circ} = \frac{1}{0}$, undefined again. Two independent methods agreeing is the sign that the answer is real, even when the answer is "no value exists."

For the wider family that secant belongs to, see cosecant, secant, and cotangent functions.

Where Does 90 Degrees Sit On The Unit Circle?

On the unit circle, every angle points to a single point whose coordinates are $(\cos\theta, \sin\theta)$. The 90-degree angle points straight up, to the top of the circle:

$$90^\circ ;\longrightarrow; (\cos 90^\circ, \sin 90^\circ) = (0, 1)$$

Secant is the reciprocal of the $x$-coordinate. Here the $x$-coordinate is $0$, so $\sec 90^\circ = \frac{1}{0}$, undefined. The point sits on the $y$-axis, where the horizontal reach is exactly nothing, and that missing horizontal reach is the whole story.

There is a right-triangle way to see the same thing, and it matches the hook. In a right triangle, $\sec\theta = \frac{\text{hypotenuse}}{\text{adjacent}}$, and as the angle grows toward $90^\circ$ the adjacent side (the ground distance under the ladder) shrinks toward zero, so the ratio grows without bound. At exactly $90^\circ$ the triangle collapses, the adjacent side is gone, and the ratio has nothing to divide by. Both pictures, the circle and the triangle, blame the same missing zero.

What Happens To The Secant Graph Near Sec 90 Degrees?

This is where "undefined" and "infinity" part ways. Watch the value of $\sec\theta$ as the angle creeps up on $90^\circ$ from each side:

Table: Secant of angles approaching 90 degrees from below and above.

Approaching from below, $\sec\theta$ races toward $+\infty$. Approaching from above, it plunges toward $-\infty$. The two sides disagree about which way to run, so no single value can sit at $90^\circ$. That disagreement is exactly why we say undefined rather than infinity: infinity would need one agreed direction, and here there are two.

On the graph of $y = \sec\theta$, this shows up as a vertical asymptote, a line the curve hugs but never touches. Secant has one of these wherever cosine hits zero, which happens at

$$\theta = 90^\circ + 180^\circ k \qquad \left(\theta = \frac{\pi}{2} + n\pi\right)$$

for any whole number $k$ (or $n$). So $90^\circ$, $270^\circ$, $-90^\circ$, and every angle a half-turn apart from them all share the same undefined status. The value at $\sec\frac{\pi}{2}$ is undefined, and so is the value at $\sec\frac{3\pi}{2}$.

What Are The Secant Values Of The Special Angles?

Sec 90 degrees is the one gap in an otherwise clean row of special-angle values. Seeing it beside its neighbours makes the gap feel less like an exception and more like the natural edge of the pattern.

Table: Cosine and secant of the first-quadrant special angles.

As the angle climbs from $0^\circ$ to $90^\circ$, cosine slides from $1$ down to $0$, so its reciprocal, secant, climbs from $1$ upward and finally runs off the scale. The related radian pages fill in the neighbours: sec π/4 equals $\sqrt{2}$, sec π/3 equals $2$, and sec 0 degrees equals $1$, the smallest secant there is.

Why Is Sec 90 Degrees Undefined?

The short answer is division by zero. The fuller answer explains why cosine, of all the ratios, is the one that vanishes here.

• Cosine measures horizontal reach. On the unit circle, $\cos\theta$ is the $x$-coordinate of the point. At $90^\circ$ the point is at the very top, $(0, 1)$, directly above the centre, so its horizontal reach is zero. See cos 90 degrees for that value on its own.

• Secant inverts that reach. Because $\sec\theta = \frac{1}{\cos\theta}$, secant asks "one divided by the horizontal reach." When the reach is zero, the question has no answer.

• Zero has no reciprocal. No number multiplied by $0$ gives $1$, so $\frac{1}{0}$ cannot name any number. This is a rule of arithmetic, not a fact about trigonometry, and secant simply inherits it.

Put together, secant is undefined at $90^\circ$ for the same reason $\frac{1}{0}$ is undefined anywhere. Trigonometry did not create the gap. It just points to where the reciprocal of cosine meets the one number that has no reciprocal.

Who Discovered The Secant Function?

Secant is younger than sine and cosine by well over a thousand years. The idea of a "reach" ratio grew out of shadow-length measurements long before anyone gave it a name.

The name itself is Renaissance-era. The Danish mathematician Thomas Fincke (1561–1656, Denmark) introduced the terms "secant" and "tangent" in his 1583 book Geometria rotundi, from the Latin secare, meaning "to cut," because the line cuts across the circle. Once secant had a name and a symbol, it took its place beside cosine as one of the six standard trigonometric functions, and its undefined points at $90^\circ$ and $270^\circ$ became a fixed feature of every table.

Where Is Sec 90 Degrees Used In The Real World?

Secant rarely gets top billing, yet it runs quietly under several everyday technologies, and its blow-up near $90^\circ$ is often the whole point.

• Map making. The Mercator projection, the map shape behind most navigation apps, stretches the world using the integral of the secant function. Near the poles the map angle approaches $90^\circ$, secant runs away, and that is exactly why Greenland looks enormous on a flat map.

• Ramps and cables. The tension along a cable or the length of a support that leans at an angle scales with the secant of that angle. As the lean approaches vertical, the required length grows sharply, which engineers plan around.

• Optics and light. In problems on refraction and lens geometry, path lengths through a slanted medium carry a secant factor, and the near-$90^\circ$ blow-up flags a grazing ray that barely enters the surface.

• Calculus. The derivative of the tangent function is $\sec^2\theta$, so secant appears the moment you study how steeply the tangent curve rises, including near its own asymptote at $90^\circ$.

One thread runs through all four: whenever a quantity depends on a shrinking horizontal reach, secant is the tool that measures how fast it grows, and $90^\circ$ is where it warns you the quantity has run off the edge.

What Are The Most Common Mistakes With Sec 90 Degrees?

These four slips account for most wrong answers on secant questions, and the first one is printed in more than one textbook.

Writing that sec 90 degrees equals infinity.

Where it slips in:

A student sees the value grow huge near $90^\circ$ and writes $\sec 90^\circ = \infty$, treating infinity as a number the function reaches.

Don't do this:

Do not record infinity as the value. From above the angle, secant runs toward $-\infty$, not $+\infty$, so the two sides never agree on a single infinite value.

The correct way:

Write $\sec 90^\circ = \text{undefined}$. If a limit is being asked for, state the one-sided behaviour separately: $\sec\theta \to +\infty$ as $\theta \to 90^\circ$ from below, and $\sec\theta \to -\infty$ from above.

Reading the calculator in the wrong angle mode.

Where it slips in:

A student types $\sec 90$ (via $1 \div \cos 90$) with the calculator set to radians, reads a number like $-2.23$, and records it as $\sec 90^\circ$.

Don't do this:

Do not trust a reciprocal-of-cosine reading without checking the mode. In radian mode, $\cos 90$ means $\cos 90$ radians, a completely different angle.

The correct way:

Switch the calculator to degree mode before evaluating $\sec 90^\circ$. A correct machine then reports an error or "undefined," because it is dividing by $\cos 90^\circ = 0$.

Taking the reciprocal of the wrong function.

Where it slips in:

A student pairs secant with sine, computing $\frac{1}{\sin 90^\circ} = \frac{1}{1} = 1$ and calling it $\sec 90^\circ$.

Don't do this:

Do not confuse secant with cosecant. Secant is the reciprocal of cosine; cosecant is the reciprocal of sine.

The correct way:

Anchor the pairing: $\sec\theta = \frac{1}{\cos\theta}$ (the "co" swaps), so $\sec 90^\circ = \frac{1}{\cos 90^\circ} = \frac{1}{0}$, undefined. The reciprocal of $\sin 90^\circ$ is $\csc 90^\circ = 1$, a different question.

Mishandling the co-function shift.

Where it slips in:

A student uses $\sec\theta = \csc(90^\circ - \theta)$ but subtracts in the wrong direction or drops the shift, landing on the wrong angle.

Don't do this:

Do not guess the co-function angle. For $\sec 90^\circ$ the shift gives $\csc(90^\circ - 90^\circ) = \csc 0^\circ$, which is $\frac{1}{\sin 0^\circ} = \frac{1}{0}$, still undefined.

The correct way:

Apply the identity exactly as written, $\sec\theta = \csc(90^\circ - \theta)$, and confirm the result matches the direct reciprocal method before trusting it.

Practice Problems On Sec 90 Degrees

Work each one, then check against the answer that follows.

1. State the value of $\sec 90^\circ$ and the reason in one line.
   (Answer: undefined, because $\cos 90^\circ = 0$ and $\sec 90^\circ = \frac{1}{0}$.)

2. Find $\sec\frac{\pi}{2}$.
   (Answer: undefined; $\frac{\pi}{2}$ is $90^\circ$, so it is the same question.)

3. Evaluate $\sec 0^\circ$ and compare it with $\sec 90^\circ$.
   (Answer: $\sec 0^\circ = \frac{1}{\cos 0^\circ} = \frac{1}{1} = 1$; $\sec 90^\circ$ is undefined. Secant runs from $1$ up to no value across the first quadrant.)

4. Is $\sec 270^\circ$ defined?
   (Answer: no; $\cos 270^\circ = 0$, so $\sec 270^\circ$ is undefined, just like $\sec 90^\circ$.)

5. Compute $\sec 60^\circ$.
   (Answer: $\frac{1}{\cos 60^\circ} = \frac{1}{1/2} = 2$.)

6. As $\theta \to 90^\circ$ from below, what does $\sec\theta$ approach?
   (Answer: $+\infty$; from above it approaches $-\infty$, which is why $\sec 90^\circ$ itself is undefined.)

Where Should You Go Next After Sec 90 Degrees?

Sec 90 degrees is one node in the reciprocal-function family, and a few natural doors open from here.

1. Secant function. See the full graph, its period, and every asymptote, so the undefined point at $90^\circ$ fits into the whole curve.

2. Cosecant, secant, and cotangent functions. Meet the three reciprocal ratios together and learn which cosine, sine, or tangent zero causes each undefined point.

3. Tan 90 degrees. The other function that is undefined at $90^\circ$, for the same division-by-zero reason.

If your child is building these foundations, a live Bhanzu trainer teaches secant starting from the "why" behind its undefined points in the Bhanzu trigonometry program.