What Is The Value Of Sin 5pi/8?
Sin 5pi/8 is $\dfrac{\sqrt{2+\sqrt{2}}}{2}$, which is approximately $0.9239$. In degrees the angle is $112.5^\circ$, because $\frac{5\pi}{8}$ radians $= \frac{5}{8}\times 180^\circ = 112.5^\circ$.
$$\sin\frac{5\pi}{8} = \sin 112.5^\circ = \frac{\sqrt{2+\sqrt{2}}}{2} \approx 0.9239$$
This is one of the angles that has a clean exact form. The number under the roots is small, but the value is genuine and constructible with a compass and straightedge, unlike an angle such as $50^\circ$, which has no simple surd. Both forms, the radian $\frac{5\pi}{8}$ and the degree $112.5^\circ$, name the same rotation, and it helps to carry both. If radians still feel unfamiliar, the primer on what a radian is is worth a detour.
How Do You Find Sin 5pi/8?
The angle $\frac{5\pi}{8} = 112.5^\circ$ lands between $90^\circ$ and $180^\circ$, so it lives in the second quadrant. Two facts about that quadrant decide everything.
The sign. By the ASTC (All, Sine, Tangent, Cosine) rule, only sine and its reciprocal are positive in the second quadrant. So $\sin\frac{5\pi}{8}$ is positive.
The reference angle. In the second quadrant the reference angle is measured from $180^\circ$, not from $90^\circ$. Here it is $180^\circ - 112.5^\circ = 67.5^\circ$.
Putting the two together reduces the problem to a first-quadrant angle:
$$\sin 112.5^\circ = +\sin 67.5^\circ$$
Now $67.5^\circ$ is half of $135^\circ$, and it is also the co-function partner of $22.5^\circ$. Either observation opens a clean exact route, which the next derivation follows in full. For the general machinery, see the trigonometric ratios of specific angles and the wider trigonometric table.
Where Does 5pi/8 Sit On The Unit Circle?
On the unit circle, the sine of an angle is the $y$-coordinate of the point where the terminal side meets the circle. For $\frac{5\pi}{8}$, that point is in the upper-left, at coordinates $(-0.3827,\ 0.9239)$.
The $y$-coordinate $0.9239$ is $\sin\frac{5\pi}{8}$, positive because the point is above the horizontal axis. The $x$-coordinate $-0.3827$ is $\cos\frac{5\pi}{8}$, negative because the point is left of the vertical axis. That single point explains both signs at once.
The unit circle also connects to a right triangle. Drop the vertical from that point to the horizontal axis, and you form a right triangle whose hypotenuse is the radius $1$, whose height is $\sin 67.5^\circ$, and whose base is $\cos 67.5^\circ$. That is the double anchor: the same value read as a coordinate on the circle and as the opposite-over-hypotenuse ratio in a triangle. The unit circle with tangent page carries the full labelled version.
How Do You Derive The Exact Value Of Sin 5pi/8?
There are two clean routes to the exact surd, and both land on the same number. That agreement is the best check that the value is right.
Route 1: supplementary reduction, then half-angle from $\cos 45^\circ$.
Start from the reduction found above, then use the co-function identity to swap sine for cosine:
$$\sin\frac{5\pi}{8} = \sin 112.5^\circ = \sin 67.5^\circ = \cos 22.5^\circ$$
Now $22.5^\circ$ is half of $45^\circ$, so apply the half-angle formula for cosine, $\cos\frac{\theta}{2} = \sqrt{\frac{1+\cos\theta}{2}}$, taking the positive root because $22.5^\circ$ is in the first quadrant:
$$\cos 22.5^\circ = \sqrt{\frac{1+\cos 45^\circ}{2}}$$
$$= \sqrt{\frac{1+\frac{\sqrt{2}}{2}}{2}}$$
$$= \sqrt{\frac{2+\sqrt{2}}{4}}$$
$$= \frac{\sqrt{2+\sqrt{2}}}{2}$$
Route 2: half-angle straight from $\frac{5\pi}{4}$.
The angle $\frac{5\pi}{8}$ is exactly half of $\frac{5\pi}{4}$, so the half-angle formula for sine applies directly. Since $\frac{5\pi}{8}$ is in the second quadrant, sine is positive, so take the positive root:
$$\sin\frac{5\pi}{8} = \sqrt{\frac{1-\cos\frac{5\pi}{4}}{2}}$$
With $\cos\frac{5\pi}{4} = -\frac{\sqrt{2}}{2}$:
$$= \sqrt{\frac{1+\frac{\sqrt{2}}{2}}{2}} = \frac{\sqrt{2+\sqrt{2}}}{2}$$
Both routes give $\dfrac{\sqrt{2+\sqrt{2}}}{2}$. Squaring the surd and taking the root numerically returns $0.9238795\ldots$, matching the decimal every calculator reports.
What Are The Sine Values Around 5pi/8?
The angle $\frac{5\pi}{8}$ belongs to the family of eighths of $\pi$, all of which have half-angle surds. Seeing them together shows why sine climbs to a peak near $\frac{\pi}{2}$ and eases back down toward $\pi$.
Table: sine, cosine, and tangent for the eighth-of-pi family (all exact where a surd exists).
Angle | Radians | $\sin$ | $\cos$ | $\tan$ |
|---|---|---|---|---|
$22.5^\circ$ | $\frac{\pi}{8}$ | $\frac{\sqrt{2-\sqrt{2}}}{2} \approx 0.3827$ | $\frac{\sqrt{2+\sqrt{2}}}{2} \approx 0.9239$ | $\approx 0.4142$ |
$67.5^\circ$ | $\frac{3\pi}{8}$ | $\frac{\sqrt{2+\sqrt{2}}}{2} \approx 0.9239$ | $\frac{\sqrt{2-\sqrt{2}}}{2} \approx 0.3827$ | $\approx 2.4142$ |
$112.5^\circ$ | $\frac{5\pi}{8}$ | $\frac{\sqrt{2+\sqrt{2}}}{2} \approx 0.9239$ | $-\frac{\sqrt{2-\sqrt{2}}}{2} \approx -0.3827$ | $\approx -2.4142$ |
$135^\circ$ | $\frac{3\pi}{4}$ | $\frac{\sqrt{2}}{2} \approx 0.7071$ | $-\frac{\sqrt{2}}{2} \approx -0.7071$ | $-1$ |
$157.5^\circ$ | $\frac{7\pi}{8}$ | $\frac{\sqrt{2-\sqrt{2}}}{2} \approx 0.3827$ | $-\frac{\sqrt{2+\sqrt{2}}}{2} \approx -0.9239$ | $\approx -0.4142$ |
Notice the symmetry: $\sin 67.5^\circ$ and $\sin 112.5^\circ$ are identical, because $112.5^\circ$ and $67.5^\circ$ are supplementary angles. The neighbouring value $\sin\frac{3\pi}{4}$ has its own page at sin 3pi/4, and radian-first practice sits at trigonometric ratios in radians.
Why Is Sin 5pi/8 Positive?
The sign follows from geometry, not from a rule to memorise.
The terminal side of $\frac{5\pi}{8}$ points into the upper-left of the plane, so its endpoint on the unit circle has a positive height.
Sine reads that height, the $y$-coordinate, so a point above the axis forces a positive sine.
Cosine reads the $x$-coordinate, which is negative here, which is why $\cos\frac{5\pi}{8}$ is negative while the sine stays positive.
Any second-quadrant angle behaves the same way, which is the geometric content of the ASTC rule.
The size, close to $0.92$, comes from $112.5^\circ$ sitting near $90^\circ$, where sine reaches its maximum of $1$. Move the angle toward $180^\circ$ and the height would shrink back toward $0$.
Who Discovered How To Compute Values Like Sin 5pi/8?
Long before calculators, astronomers needed the length of a chord for every angle they could measure in the sky, and the half-angle trick is exactly how they filled the gaps between the angles they already knew.
Two other figures shaped this lineage:
Hipparchus of Nicaea (c. 190–120 BCE, Greece) is credited with the first known table of chords, the direct ancestor of the sine table, which is why he is often called the founder of trigonometry.
Aryabhata (476–550 CE, India) tabulated the half-chord, the quantity he called jya, whose name travelled through Arabic into the Latin sinus and gave us the modern word "sine."
Where Is Sin 5pi/8 Used In The Real World?
Values like $\sin\frac{5\pi}{8}$ appear whenever something rotates or oscillates past the quarter turn.
Waves and sound: the instantaneous height of a sound or water wave at a phase of $112.5^\circ$ is proportional to $\sin\frac{5\pi}{8}$, still near its crest.
Alternating current: an AC voltage a little past its peak, at $112.5^\circ$ of the cycle, sits at $0.9239$ of its maximum, a figure power engineers read straight off the sine curve.
Circular motion and orbits: the vertical position of a point moving around a circle at angle $\frac{5\pi}{8}$ is exactly this sine value, which is how animators and astronomers alike track height.
Computer graphics: rotating a vertex by $112.5^\circ$ multiplies its coordinates by sines and cosines of that angle, so this exact value is baked into rotation matrices.
Navigation and surveying: bearings past due-east use second-quadrant sines to resolve how far north a heading carries you.
Across all of them, the reason is the same: sine measures a height on a turning circle, and $\frac{5\pi}{8}$ is simply one position on that turn.
What Are The Most Common Mistakes With Sin 5pi/8?
These four errors account for most lost marks on second-quadrant values, and each has a clean fix.
Making the sine negative.
Where it slips in:
A student sees that $\frac{5\pi}{8}$ is past $90^\circ$ and assumes, as with cosine or tangent there, that the value must be negative.
Don't do this:
Do not attach a minus sign. Sine is positive throughout the second quadrant.
The correct way:
Apply ASTC. In the second quadrant sine is positive, so $\sin\frac{5\pi}{8} = +\dfrac{\sqrt{2+\sqrt{2}}}{2}$. Only cosine and tangent turn negative there.
Reading the reference angle from the wrong axis.
Where it slips in:
A student computes the reference angle as $112.5^\circ - 90^\circ = 22.5^\circ$ instead of $180^\circ - 112.5^\circ = 67.5^\circ$.
Don't do this:
Do not subtract from $90^\circ$ in the second quadrant.
The correct way:
In the second quadrant, reference angle $= 180^\circ - \theta$. Here that is $67.5^\circ$, so $\sin\frac{5\pi}{8} = \sin 67.5^\circ$, not $\sin 22.5^\circ$.
Leaving the calculator in the wrong mode.
Where it slips in:
A student types $\sin(5\pi/8)$ with the calculator set to degrees, and gets the sine of about $1.96^\circ$ instead.
Don't do this:
Do not mix the forms. An entry in radians needs radian mode; an entry of $112.5$ needs degree mode.
The correct way:
Match the mode to the input. Enter $112.5$ in degree mode, or $5\pi/8$ in radian mode. Both return $0.9239$.
Confusing the co-function.
Where it slips in:
While swapping $\sin 67.5^\circ$ for a cosine, a student writes $\cos 67.5^\circ$ instead of $\cos 22.5^\circ$.
Don't do this:
Do not keep the same angle when switching function.
The correct way:
The co-function pairs an angle with its complement: $\sin 67.5^\circ = \cos(90^\circ - 67.5^\circ) = \cos 22.5^\circ$. The angle changes to its complement when the function changes.
Practice Problems On Sin 5pi/8
Work each one, then check against the answer. Values are exact where a surd exists.
State $\sin\frac{5\pi}{8}$ in both exact and decimal form.
(Answer: $\dfrac{\sqrt{2+\sqrt{2}}}{2} \approx 0.9239$.)Convert $\frac{5\pi}{8}$ radians to degrees.
(Answer: $112.5^\circ$.)Find $\cos\frac{5\pi}{8}$ exactly.
(Answer: $-\dfrac{\sqrt{2-\sqrt{2}}}{2} \approx -0.3827$; cosine is negative in the second quadrant.)Use the identity $\sin(\pi - x) = \sin x$ to explain why $\sin\frac{5\pi}{8} = \sin\frac{3\pi}{8}$.
(Answer: $\pi - \frac{5\pi}{8} = \frac{3\pi}{8}$, so the supplementary angles share the same sine, $0.9239$.)Compute $\tan\frac{5\pi}{8}$ using your answers to 1 and 3.
(Answer: $\dfrac{\sin}{\cos} = \dfrac{0.9239}{-0.3827} \approx -2.4142$.)Without a calculator, decide whether $\sin\frac{5\pi}{8}$ or $\sin\frac{3\pi}{4}$ is larger.
(Answer: $\sin\frac{5\pi}{8} \approx 0.9239$ is larger than $\sin\frac{3\pi}{4} \approx 0.7071$, since $112.5^\circ$ is closer to the peak at $90^\circ$.)
Where Should You Go Next After Sin 5pi/8?
A few natural doors open from this value.
Half-angle formula. The single tool behind this value, useful for every angle that is half of a known one.
Sine function. See how these point values join into the full sine wave, crest and all.
Sin, cos, tan. Revisit the three core ratios and how quadrant signs flow between them.
If your child is building this foundation, a live Bhanzu trainer teaches values like $\sin\frac{5\pi}{8}$ starting from the unit circle and the "why" behind each sign, in the Bhanzu trigonometry program.
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