Sec pi/4 : Exact Value √2 and How to Find It

What Does Sec π/4 Mean?

Secant is one of the three reciprocal trigonometric ratios. For any angle, $\sec\theta = \frac{1}{\cos\theta}$, the reciprocal of the cosine, as set out in the reciprocal identities.

In a right triangle, cosine is adjacent over hypotenuse, so secant is the flip: hypotenuse over adjacent. So $\sec\frac{\pi}{4}$ asks how many times longer the hypotenuse is than the side next to the $45^\circ$ angle.

On the unit circle - radius $1$, centred at the origin - cosine is the $x$-coordinate where the angle's radius meets the circle, and secant is the reciprocal of that $x$-coordinate. The angle $\frac{\pi}{4}$ sits in the first quadrant, the quarter of the plane where both coordinates are positive, so its secant is positive.

What Is the Value of Sec π/4?

Sec π/4 equals $\sqrt{2}$, approximately $1.4142$. It is an exact value, because $45^\circ$ is a standard angle.

Quick Answer

Result: $\sec\dfrac{\pi}{4} = \sqrt{2}$ Decimal: $\approx 1.4142$ Reason: $\sec\dfrac{\pi}{4} = \dfrac{1}{\cos\frac{\pi}{4}} = \dfrac{1}{\frac{\sqrt{2}}{2}} = \dfrac{2}{\sqrt{2}} = \sqrt{2}$ In degrees: $\sec 45^\circ = \sqrt{2}$ (same angle, different unit) Methods shown: reciprocal of cosine · 45-45-90 triangle · unit-circle $x$-coordinate

Sec π/4 radians names the same angle as sec 45°, so both equal $\sqrt{2}$. The radian form is the one that turns up on the unit circle and in integrals, so it earns its own page.

Where Does Sec π/4 Show Up?

A $45^\circ$ angle is the diagonal of a square, and secant of $45^\circ$ is the factor that turns a horizontal run into that diagonal length. Cut a square brace or a staircase stringer at $45^\circ$ and the sloped piece is the base run multiplied by $\sqrt{2}$, which is why $\sqrt{2}$ shows up in so many construction cuts.

The value also lands in signal work: the root-mean-square relationship for an alternating current uses the same $\frac{1}{\sqrt{2}}$ and its reciprocal $\sqrt{2}$. Because $\frac{\pi}{4}$ splits a right angle exactly in half, its secant is a benchmark number worth carrying in your head.

Standard-Angle Secant Reference Table

Secant is the reciprocal of cosine, so each secant value is $1$ divided by the matching cosine. Here are the first-quadrant standard angles in both units.

Reading down, $\sec\frac{\pi}{4} = \sqrt{2}$ sits between $\sec\frac{\pi}{6}$ and $\sec\frac{\pi}{3} = 2$. The rationalised form $\sqrt{2}$ is the standard way to write it; $\frac{2}{\sqrt{2}}$ is the same number before the denominator is cleared.

How Do You Find the Exact Value of Sec π/4?

Three routes all give $\sqrt{2}$; holding more than one keeps the value from being something you have to trust on faith.

Method 1: Reciprocal of cosine.

Start from the known cosine, flip it, and rationalise the denominator.

$$\sec\frac{\pi}{4} = \frac{1}{\cos\frac{\pi}{4}} = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}} = \frac{2}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{2\sqrt{2}}{2} = \sqrt{2}$$

Method 2: The 45-45-90 triangle.

Take a right triangle with both legs equal to $1$. By the Pythagorean theorem the hypotenuse is $\sqrt{1^2 + 1^2} = \sqrt{2}$, and the side adjacent to either $45^\circ$ angle is $1$.

$$\sec 45^\circ = \frac{\text{hypotenuse}}{\text{adjacent}} = \frac{\sqrt{2}}{1} = \sqrt{2}$$

Method 3: The unit circle.

Rotate the radius $45^\circ$ from the positive $x$-axis. The tip lands at $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$, so the $x$-coordinate is $\frac{\sqrt{2}}{2}$.

$$\sec\frac{\pi}{4} = \frac{1}{x\text{-coordinate}} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}$$

The three agree because the unit circle is the 45-45-90 triangle scaled so the hypotenuse is $1$. The full graph these values sit on lives on the secant function page.

Examples of Sec π/4

Example 1

Evaluate $\sec^2\frac{\pi}{4}$.

$$\sec^2\frac{\pi}{4} = \left(\sqrt{2}\right)^2 = 2$$

Example 2

A student writes $\sec\frac{\pi}{4} = \frac{\sqrt{2}}{2}$. What went wrong?

Wrong attempt. The reasoning is "cos of $\frac{\pi}{4}$ is $\frac{\sqrt{2}}{2}$, so the secant is $\frac{\sqrt{2}}{2}$ too."

That writes down the cosine value instead of its reciprocal. Check it against size: secant of a first-quadrant angle is at least $1$, but $\frac{\sqrt{2}}{2} \approx 0.71$ is less than $1$, so it cannot be the secant.

Correct. $\sec\frac{\pi}{4} = \frac{1}{\cos\frac{\pi}{4}} = \frac{2}{\sqrt{2}} = \sqrt{2} \approx 1.41$. The reciprocal flips a number below $1$ to one above it.

Example 3

A right triangle has a $45^\circ$ angle and an adjacent side of $6$ cm. Find the hypotenuse.

$$\sec 45^\circ = \frac{\text{hypotenuse}}{6} \quad\Rightarrow\quad \text{hypotenuse} = 6 \times \sqrt{2} = 6\sqrt{2} \approx 8.49 \text{ cm}$$

The hypotenuse is $\sqrt{2}$ times the adjacent side, the geometric meaning of $\sec 45^\circ = \sqrt{2}$.

Example 4

Verify that $\sec\frac{\pi}{4} \times \cos\frac{\pi}{4} = 1$.

$$\sec\frac{\pi}{4} \times \cos\frac{\pi}{4} = \sqrt{2} \times \frac{\sqrt{2}}{2} = \frac{2}{2} = 1$$

A value times its reciprocal is $1$, as the reciprocal relationship requires.

Example 5

Express $\sec\frac{\pi}{4}$ in degrees and confirm the value.

Since $\frac{\pi}{4} = \frac{\pi}{4} \times \frac{180^\circ}{\pi} = 45^\circ$, we have $\sec\frac{\pi}{4} = \sec 45^\circ$. Both rest on $\cos 45^\circ = \frac{\sqrt{2}}{2}$, so both equal $\sqrt{2}$; only the unit label changes.

Where Students Trip Up on Sec π/4

Mistake 1: Leaving the answer as √2/2 instead of √2

Where it slips in: Copying the cosine value $\frac{\sqrt{2}}{2}$ straight across as the secant.

Don't do this: $\sec\frac{\pi}{4} = \frac{\sqrt{2}}{2}$.

The correct way: Take the reciprocal, $\frac{1}{\sqrt{2}/2} = \frac{2}{\sqrt{2}} = \sqrt{2}$. The first-instinct error is treating secant as a copy of cosine; the fix is to write $\frac{1}{\cos\theta}$ before reading any value.

Mistake 2: Stopping at 2/√2 without rationalising

Where it slips in: Getting $\frac{2}{\sqrt{2}}$ and leaving it, unsure whether it equals $\sqrt{2}$.

Don't do this: Reporting $\frac{2}{\sqrt{2}}$ as the final form on a problem that expects the standard value.

The correct way: Multiply top and bottom by $\sqrt{2}$: $\frac{2}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{2\sqrt{2}}{2} = \sqrt{2}$. The memoriser who never rationalises leaves half-simplified answers; $\sqrt{2}$ is the rationalised standard form.

Mistake 3: Forgetting the calculator's angle mode

Where it slips in: A calculator left in radian mode when you type $45$, or in degree mode when you type the decimal for $\frac{\pi}{4}$.

Don't do this: Trusting the screen without checking the mode.

The correct way: Match the mode to the input; a true $45^\circ$ returns $1.4142\ldots$, and anything far from that signals a mode mismatch.

Key Takeaways

• Sec π/4 equals $\sqrt{2}$, about $1.4142$, because $\sec\frac{\pi}{4} = \frac{1}{\cos\frac{\pi}{4}} = \frac{2}{\sqrt{2}} = \sqrt{2}$.

• The 45-45-90 triangle gives it as hypotenuse over adjacent, $\frac{\sqrt{2}}{1}$; the unit circle gives it as the reciprocal of the $x$-coordinate $\frac{\sqrt{2}}{2}$.

• Sec π/4 and $\sec 45^\circ$ are the same value — the unit changes, the number does not.

• The most common slips are copying $\frac{\sqrt{2}}{2}$ and stopping at $\frac{2}{\sqrt{2}}$; rationalised, the answer is $\sqrt{2}$.

To take sec π/4 and the special angles further with a teacher, explore Bhanzu's trigonometry tutor sessions, a high school math tutor for exam practice, or live math classes online with peers from 20+ countries.

Practice Sec π/4 Before Moving On

1. Evaluate $\sec\frac{\pi}{4} + \sec\frac{\pi}{3}$ and give the exact form.

2. A staircase stringer runs at $45^\circ$ over a horizontal run of $2$ m. Use $\sec 45^\circ$ to find its length.

3. Show that $\sec^2\frac{\pi}{4} - 1 = 1$ and name the identity that predicts this.

Want a live Bhanzu trainer to work through the reciprocal ratios with you? Book a free demo class.

Read More

• Sec π/3 — the neighbouring reciprocal value, equal to 2.

• Sec π/2 — the angle where secant runs off to undefined.

• Cos 30 degrees — another exact standard-angle value with a radical form.

• Cosine function — the parent function secant is built from.

• Sin cos tan — the three primary ratios behind all the reciprocals.