Cos pi/8: Exact Value & How To Find It

What Is The Value Of Cos pi/8?

Cos pi/8 equals $\dfrac{\sqrt{2+\sqrt{2}}}{2}$, which is approximately $0.9239$ to four decimal places. The angle is written two ways that mean exactly the same thing: $\frac{\pi}{8}$ in radians and $22.5^\circ$ in degrees.

$$\cos\frac{\pi}{8} = \cos 22.5^\circ = \frac{\sqrt{2+\sqrt{2}}}{2} \approx 0.9239$$

The value is close to $1$ because $22.5^\circ$ is a small angle, and the cosine of a small angle is near its maximum of $1$. The two ways of naming the angle are linked by the fact that $\pi$ radians equals $180^\circ$, so $\frac{\pi}{8} = \frac{180^\circ}{8} = 22.5^\circ$. If the radian idea is new, the page on what a radian is sets it up from scratch.

How Do You Find Cos pi/8?

Because $22.5^\circ$ is not one of the standard angles ($0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ$), you cannot read it off the trigonometric table directly. The trick is that $22.5^\circ$ is exactly half of $45^\circ$, an angle you already know. That opens the half-angle formula.

The half-angle formula for cosine is:

$$\cos\frac{\theta}{2} = \pm\sqrt{\frac{1+\cos\theta}{2}}$$

The $\pm$ sign is decided by the quadrant the half-angle lands in. Here $\frac{\theta}{2} = \frac{\pi}{8} = 22.5^\circ$ sits in the first quadrant, where cosine is positive, so you take the plus sign. Setting $\theta = \frac{\pi}{4} = 45^\circ$ and using $\cos\frac{\pi}{4} = \frac{\sqrt{2}}{2}$:

$$\cos\frac{\pi}{8} = \sqrt{\frac{1+\cos\frac{\pi}{4}}{2}} = \sqrt{\frac{1+\frac{\sqrt{2}}{2}}{2}}$$

Now clear the inner fraction by writing $1 = \frac{2}{2}$:

$$= \sqrt{\frac{\frac{2+\sqrt{2}}{2}}{2}} = \sqrt{\frac{2+\sqrt{2}}{4}}$$

The denominator $4$ is a perfect square, so its root comes out as $2$:

$$= \frac{\sqrt{2+\sqrt{2}}}{2} \approx 0.9239$$

That nested square root is the exact value. It is not a "messier" answer than $\frac{\sqrt{2}}{2}$, it is the same kind of surd, just one layer deeper because the angle was halved once more.

Where Does pi/8 Sit On The Unit Circle?

On the unit circle, an angle is measured anticlockwise from the positive $x$-axis, and the point where the angle's ray meets the circle has coordinates $(\cos\theta, \sin\theta)$. So the cosine is simply the $x$-coordinate of that point.

The angle $\frac{\pi}{8} = 22.5^\circ$ lands in the first quadrant, a little above the $x$-axis and well to the right. Its point on the circle is $\left(\frac{\sqrt{2+\sqrt{2}}}{2}, \frac{\sqrt{2-\sqrt{2}}}{2}\right)$, roughly $(0.9239, 0.3827)$. Because the point sits far to the right, its $x$-coordinate is large and positive, which is exactly why cos pi/8 is close to $1$.

How Does Cos pi/8 Come From A Right Triangle?

The unit circle and the right triangle are two views of the same value, and seeing both stops cosine from feeling like two unrelated ideas. Drop that vertical dashed line from the circle point down to the $x$-axis, and you have a right triangle inside the circle.

• The hypotenuse is the radius, length $1$.

• The adjacent side (along the $x$-axis) has length $\cos\frac{\pi}{8}$.

• The opposite side (the vertical drop) has length $\sin\frac{\pi}{8}$.

By the classic ratio, $\cos\frac{\pi}{8} = \dfrac{\text{adjacent}}{\text{hypotenuse}} = \dfrac{\cos\frac{\pi}{8}}{1}$, which is the same $0.9239$. If you build a right triangle with a $22.5^\circ$ angle and a hypotenuse of $10\text{ cm}$, the side next to the angle measures $10 \times 0.9239 = 9.239\text{ cm}$. For a refresher on these three ratios, see sin cos tan and the wider set of trigonometric ratios of specific angles.

What Are The Related Values Around pi/8?

The half-angle method gives all three main ratios at $\frac{\pi}{8}$, and it is worth seeing them beside the parent angle $\frac{\pi}{4}$ they were halved from.

Table: Exact values for pi/8 and the neighbouring special angles, in degrees and radians.

A useful check sits in that table. Since $22.5^\circ$ is smaller than $30^\circ$ and $45^\circ$, its cosine ($0.9239$) is the largest of the three, because cosine shrinks as the angle grows from $0^\circ$ to $90^\circ$. The related cos 15 degrees and cos 45 degrees pages walk the same half-angle idea for their own angles.

Why Is Cos pi/8 Equal To √(2+√2)/2?

The nested-root shape is not an accident of algebra. It is the signature of an angle made by halving.

• Halving an angle deepens the surd. The known angle $\frac{\pi}{4}$ already carries one square root, $\frac{\sqrt{2}}{2}$. The half-angle formula wraps that value inside a second square root, so $\frac{\pi}{8}$ carries a root inside a root. Halve again to reach $\frac{\pi}{16}$, and a third layer appears.

• The plus sign records the quadrant. The formula offers $\pm$, and the geometry chooses. Since $\frac{\pi}{8}$ lands in the first quadrant, where every $x$-coordinate is positive, cosine must be positive, so the answer keeps the plus.

• The size matches the picture. A value of $0.9239$ means the point on the unit circle is almost as far right as it can be. That fits an angle of only $22.5^\circ$, a small tilt away from the $x$-axis.

Put together, the exact value is the algebra faithfully recording three facts at once: which angle was halved, which quadrant it lives in, and how close to the axis it sits.

Who Discovered How To Halve An Angle?

Long before the half-angle formula was written in modern symbols, astronomers needed the cosine of awkward angles to predict where planets would be, and they built the halving trick to get there.

Two other figures shaped the same story:

• Hipparchus of Nicaea (c. 190 – c. 120 BCE, Greece) built the first known table of chords, the work Ptolemy later extended, and is often called the founder of trigonometry.

• Aryabhata (476 – 550 CE, India) compiled an early table of half-chords, the jya values that gave us the very word "sine," and computed them at fine intervals for astronomy.

Where Is Cos pi/8 Used In The Real World?

The $22.5^\circ$ angle behind cos pi/8 is the natural angle of anything divided into eight, and it shows up wherever eightfold symmetry does.

• Carpentry and manufacturing: an octagonal table, gazebo, or picture frame needs eight mitre cuts, and the geometry of each corner depends on the $22.5^\circ$ half-angle, with $\cos 22.5^\circ$ setting the width a board covers.

• Signal processing: the Fast Fourier Transform, which powers audio and image compression, repeatedly uses the cosines of angles like $\frac{\pi}{8}$ as it splits a signal into eight or sixteen parts.

• Antenna and radar arrays: phased arrays steer a beam by shifting signals across elements spaced at fractions of a wave, and $22.5^\circ$ phase steps are a common design choice.

• Computer graphics: rotating a sprite or texture by a smooth $22.5^\circ$ needs $\cos 22.5^\circ$ inside the rotation matrix that repositions every pixel.

• Navigation: the 16-point compass rose divides the circle into $22.5^\circ$ sectors, so each named heading, such as "north-northeast," sits one cos pi/8 step from its neighbour.

One small angle, halved out of $45^\circ$, quietly shapes furniture, sound, radio, screens, and the compass.

What Are The Most Common Mistakes With Cos pi/8?

These four errors account for most lost marks on half-angle values, and they were the exact slips surfaced when searching how students handle $\cos 22.5^\circ$.

Choosing the wrong sign in the half-angle formula.

Where it slips in:

A student writes $\cos\frac{\pi}{8} = \pm\sqrt{\frac{1+\cos\frac{\pi}{4}}{2}}$ and leaves the $\pm$, or picks the minus sign out of habit.

Don't do this:

Do not carry the $\pm$ into the final answer, and do not guess the sign.

The correct way:

Check the quadrant of the half-angle first. Here $\frac{\pi}{8} = 22.5^\circ$ is in the first quadrant, cosine is positive there, so the sign is plus and the answer is $+\frac{\sqrt{2+\sqrt{2}}}{2}$.

Reading the calculator in the wrong angle mode.

Where it slips in:

A student types $\cos(\pi/8)$ with the calculator set to degrees, or types $\cos(22.5)$ with it set to radians, and gets a value that is not $0.9239$.

Don't do this:

Do not enter an angle before confirming the MODE. Typing $\pi/8 \approx 0.3927$ in degree mode returns about $0.99998$, not the intended value.

The correct way:

Match the mode to the angle. Use radian mode for $\frac{\pi}{8}$ and degree mode for $22.5^\circ$. Both should return $0.9239$.

Halving the value instead of the angle.

Where it slips in:

A student reasons that since $22.5^\circ$ is half of $45^\circ$, the cosine must be half of $\cos 45^\circ$, giving $\frac{1}{2}\cdot\frac{\sqrt{2}}{2} \approx 0.3536$.

Don't do this:

Do not halve the cosine when you halve the angle. Cosine is not a straight-line (linear) function of the angle.

The correct way:

Feed the angle through the half-angle formula. Halving $45^\circ$ to $22.5^\circ$ raises the cosine from $0.7071$ toward $0.9239$, it does not cut it in half.

Confusing the co-function relationship.

Where it slips in:

A student sets $\cos\frac{\pi}{8}$ equal to $\sin\frac{\pi}{8}$, or mixes up which complementary angle pairs with it.

Don't do this:

Do not pair cosine with the same angle's sine. The co-function link uses the complementary angle.

The correct way:

Use $\cos\theta = \sin\left(\frac{\pi}{2}-\theta\right)$, so $\cos\frac{\pi}{8} = \sin\frac{3\pi}{8} = \sin 67.5^\circ \approx 0.9239$. The cofunction identities page sets out the full rule.

Practice Problems On Cos pi/8

Work each one, then check against the answer.

1. Convert $\frac{\pi}{8}$ radians to degrees.
   (Answer: $\frac{180^\circ}{8} = 22.5^\circ$.)

2. State the exact value of $\cos\frac{\pi}{8}$.
   (Answer: $\frac{\sqrt{2+\sqrt{2}}}{2} \approx 0.9239$.)

3. A right triangle has a $22.5^\circ$ angle and a hypotenuse of $8\text{ cm}$. Find the side adjacent to the angle.
   (Answer: $8\cos 22.5^\circ = 8 \times 0.9239 \approx 7.39\text{ cm}$.)

4. Use the co-function identity to write $\cos\frac{\pi}{8}$ as a sine.
   (Answer: $\cos\frac{\pi}{8} = \sin\frac{3\pi}{8} = \sin 67.5^\circ$.)

5. Given $\cos\frac{\pi}{8} = \frac{\sqrt{2+\sqrt{2}}}{2}$, verify that $\sin\frac{\pi}{8} = \frac{\sqrt{2-\sqrt{2}}}{2}$ using $\sin^2\theta + \cos^2\theta = 1$.
   (Answer: $\cos^2\frac{\pi}{8} = \frac{2+\sqrt{2}}{4}$, so $\sin^2\frac{\pi}{8} = 1 - \frac{2+\sqrt{2}}{4} = \frac{2-\sqrt{2}}{4}$, giving $\sin\frac{\pi}{8} = \frac{\sqrt{2-\sqrt{2}}}{2}$.)

6. Is $\cos\frac{\pi}{8}$ larger or smaller than $\cos\frac{\pi}{6}$, and why?
   (Answer: Larger. $\frac{\pi}{8} = 22.5^\circ$ is a smaller angle than $\frac{\pi}{6} = 30^\circ$, and cosine decreases from $0^\circ$ to $90^\circ$, so $0.9239 > 0.8660$.)

Where Should You Go Next After Cos pi/8?

Cos pi/8 is one worked case of a much wider toolkit, and a few natural doors open from here.

1. Half angle formula. The single method behind this value, with the full set of sine, cosine, and tangent versions and when to use each.

2. Sum and difference identities. The other route to non-standard angles, and the source of values like $\cos 15^\circ$ from $45^\circ - 30^\circ$.

3. Unit circle with tangent. See where every angle, including $\frac{\pi}{8}$, lands and how its coordinates give sine, cosine, and tangent at a glance.

If your child is building these foundations, a live Bhanzu trainer teaches half-angle values starting from the "why" (the unit circle and the geometry of halving) in the Bhanzu trigonometry program.