Sec 45 Degrees: Exact Value √2 And Steps

What Is The Value Of Sec 45 Degrees?

The value of Sec 45 Degrees is $\sqrt{2}$, which is about 1.4142 to four decimal places. Written in full, $\sec 45^\circ = \sqrt{2} \approx 1.4142$, and because $45^\circ = \frac{\pi}{4}$ radians, the same fact reads $\sec\frac{\pi}{4} = \sqrt{2}$.

Secant is the reciprocal of cosine, so $\sec 45^\circ = \dfrac{1}{\cos 45^\circ}$. Since $45^\circ$ lands in the first quadrant, where every ratio is positive, the answer carries a plus sign. For the function itself, see secant function and the wider family in cosecant, secant, and cotangent functions.

Table: Sec 45 Degrees at a glance, in both degree and radian form.

How Do You Find Sec 45 Degrees?

Two routes reach the same value: the reciprocal identity and the right triangle. Both anchor the number so it never floats free of its meaning.

Method 1: the reciprocal of cosine. Start from the definition and substitute the known cosine.

$$\sec 45^\circ = \frac{1}{\cos 45^\circ}$$

$$\cos 45^\circ = \frac{\sqrt{2}}{2}$$

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

$$\frac{2}{\sqrt{2}} = \frac{2}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{2\sqrt{2}}{2} = \sqrt{2}$$

The last line rationalises the denominator, turning $\frac{2}{\sqrt{2}}$ into the clean surd $\sqrt{2}$. The reciprocal identity behind this step is set out in reciprocal identities, and the cosine it rests on lives at cos 45 degrees.

Method 2: the 45-45-90 right triangle. Take a right triangle with both legs equal to $1$. By the Pythagorean theorem the hypotenuse is $\sqrt{1^2 + 1^2} = \sqrt{2}$. Secant is hypotenuse over adjacent.

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

Both routes give $\sqrt{2}$, so the value holds whether you think in reciprocals or in triangle sides.

Where Does 45° Sit On The Unit Circle?

On the unit circle, the $45^\circ$ point has coordinates $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$. The x-coordinate is $\cos 45^\circ$, so the secant is one divided by that x-coordinate.

$$\sec 45^\circ = \frac{1}{x} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}$$

The point sits in the first quadrant, above and to the right of the centre, which is why the value is positive. For a fuller tour of the circle, see unit circle with tangent and the radian idea at what is a radian.

Table: Secant across the first-quadrant special angles, in degrees and radians.

The full set of values lives in the trigonometric table and the reasoning behind each entry in trigonometric ratios of specific angles.

Why Is Sec 45 Degrees Equal To √2?

The value is not arbitrary. It falls out of what happens at exactly $45^\circ$, from three angles of view.

• The triangle is balanced. At $45^\circ$ the two legs of a right triangle are equal, so the hypotenuse is $\sqrt{2}$ times a leg, and hypotenuse-over-adjacent is exactly $\sqrt{2}$.

• The x-coordinate is $\frac{\sqrt{2}}{2}$. On the unit circle the $45^\circ$ point has $x = \frac{\sqrt{2}}{2}$, and secant is $\frac{1}{x}$, which rationalises to $\sqrt{2}$.

• The quadrant sets the sign. The point sits in the first quadrant, where all six ratios are positive, so no minus sign appears.

One extra fact ties it together: $45^\circ$ is self-complementary, meaning it is its own complement ($90^\circ - 45^\circ = 45^\circ$). Because secant and cosecant are cofunctions, this forces $\sec 45^\circ = \csc 45^\circ = \sqrt{2}$. The cofunction rule behind that is set out in cofunction identities.

Who Discovered The Values Behind Sec 45 Degrees?

No single person discovered $\sec 45^\circ$. The table of angle values it belongs to was built up over roughly sixteen centuries, by astronomers who needed to predict where the sky would be.

Two later figures shaped the values into the form students meet now:

• Aryabhata (476–550 CE, India) compiled an early table of half-chords, the jya values that became our sine, in his work the Aryabhatiya around 500 CE.

• Thomas Fincke (1561–1656, Denmark) introduced the terms secant and tangent in his 1583 book Geometria rotundi, giving the ratio behind $\sec 45^\circ$ the name it still carries.

Where Is Sec 45 Degrees Used In The Real World?

A $45^\circ$ angle is one of the most common slopes in everyday design, so its secant shows up in practical measurement more often than students expect.

• Ramps and roofs: a slope built at $45^\circ$ has a diagonal length equal to $\sqrt{2}$ times its horizontal run, the secant relationship, which sets material lengths for roofers and ramp builders.

• Photography and film: a camera framed at a $45^\circ$ tilt covers a diagonal distance scaled by $\sec 45^\circ$, useful when planning how much of a scene fits the frame.

• Navigation and surveying: sighting a landmark at $45^\circ$ makes the line-of-sight distance $\sqrt{2}$ times the ground distance, a quick field estimate.

• Engineering and physics: forces resolved along a $45^\circ$ brace or an inclined plane use the same $\sqrt{2}$ scaling to convert between along-slope and horizontal measurements.

The pattern is always the same: whenever a right triangle holds a $45^\circ$ angle, the long side is $\sqrt{2}$ times the base, and that is exactly what the secant reports.

What Are The Most Common Mistakes With Sec 45 Degrees?

These three errors account for most wrong answers on secant of a special angle. Each is easy to avoid once you see where it slips in.

Leaving the calculator in the wrong mode.

Where it slips in:

A student types the angle expecting degrees, but the calculator is set to radians, so it evaluates the secant of $45$ radians instead of $45^\circ$.

Don't do this:

Do not trust the number before checking the mode indicator. The two answers are completely different.

The correct way:

Set the calculator to degrees for $45^\circ$, or convert first: $45^\circ = \frac{\pi}{4}$ radians, then evaluate. Either way, confirm the mode before reading the result.

Confusing secant with the inverse cosine.

Where it slips in:

A student reads $\sec 45^\circ$ as $\cos^{-1} 45^\circ$ and reaches for the arccosine button, treating the $-1$ notation and the reciprocal as the same thing.

Don't do this:

Do not confuse the reciprocal with the inverse function. Secant is $\frac{1}{\cos}$; arccosine is the angle whose cosine you are given.

The correct way:

Compute $\sec 45^\circ$ as $\dfrac{1}{\cos 45^\circ} = \dfrac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}$. The reciprocal rule is spelled out in reciprocal identities.

Stopping at $\frac{2}{\sqrt{2}}$ without rationalising.

Where it slips in:

A student reaches $\dfrac{2}{\sqrt{2}}$ and writes that as the final answer, or misreads it as $2$.

Don't do this:

Do not leave a root in the denominator. It hides that the value is the familiar $\sqrt{2}$.

The correct way:

Multiply top and bottom by $\sqrt{2}$: $\dfrac{2}{\sqrt{2}} \times \dfrac{\sqrt{2}}{\sqrt{2}} = \dfrac{2\sqrt{2}}{2} = \sqrt{2} \approx 1.4142$.

Practice Problems On Sec 45 Degrees

Work each one, then check the answer beside it.

1. Evaluate $\sec 45^\circ$.
   (Answer: $\sqrt{2} \approx 1.4142$.)

2. Show that $\sec 45^\circ = \csc 45^\circ$.
   (Answer: both equal $\sqrt{2}$, since $\sin 45^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2}$.)

3. Simplify $\sec^2 45^\circ$.
   (Answer: $(\sqrt{2})^2 = 2$.)

4. Given $\cos\theta = \frac{\sqrt{2}}{2}$ with $\theta$ in the first quadrant, find $\sec\theta$.
   (Answer: $\sqrt{2}$.)

5. Evaluate $\sec\frac{\pi}{4} + \cos 45^\circ$.
   (Answer: $\sqrt{2} + \frac{\sqrt{2}}{2} = \frac{3\sqrt{2}}{2} \approx 2.1213$.)

6. A right isosceles triangle has legs of $5$ cm. Find the secant of one base angle.
   (Answer: the base angle is $45^\circ$, so $\sec 45^\circ = \frac{5\sqrt{2}}{5} = \sqrt{2} \approx 1.4142$.)

Where Should You Go Next After Sec 45 Degrees?

Once $\sec 45^\circ = \sqrt{2}$ feels solid, several natural doors open from here.

1. Cos 45 degrees. The cosine that secant is built from, worked through from the same triangle and circle.

2. Sec π/3. The next special-angle secant, where the value jumps to a clean $2$.

3. Trigonometric ratios of specific angles. The whole grid of $0^\circ$ to $90^\circ$ values, so no single result stands alone.

If your child is building these foundations, a live Bhanzu trainer teaches special-angle values starting from the triangle and the unit circle in the Bhanzu trigonometry program.