Sin 24 Degrees: Value, Radians & Unit Circle

What Is The Value Of Sin 24 Degrees?

Sin 24 degrees is approximately $0.4067$, or $0.4067366$ to seven places. Written as a mapping, $\sin 24^\circ \approx 0.4067$. In radians the angle is $24^\circ = \dfrac{2\pi}{15} \approx 0.4189$, so the same fact reads $\sin\dfrac{2\pi}{15} \approx 0.4067$.

Two quick facts fix the value in place before any calculation:

• Sign: $24^\circ$ lies in the first quadrant (between $0^\circ$ and $90^\circ$), where every trigonometric ratio is positive. So $\sin 24^\circ$ is positive.

• Size: $24^\circ$ sits between $0^\circ$ (where sine is $0$) and $30^\circ$ (where sine is $0.5$), so its sine must land between those, and $0.4067$ does.

Unlike $30^\circ$, $45^\circ$, or $60^\circ$, the angle $24^\circ$ has no clean closed form such as $\tfrac{1}{2}$ or $\tfrac{\sqrt{3}}{2}$. It does have a genuine exact value, covered below, but that form is a nested radical you would never use by hand. For every practical purpose the working value is the decimal $0.4067$.

Table: Sin 24 degrees at a glance.

How Do You Find Sin 24 Degrees?

Because $24^\circ$ is not one of the standard angles on the trigonometric table, you reach it by writing it in terms of angles you already know. The cleanest route is the difference $24^\circ = 60^\circ - 36^\circ$, since both $\sin 60^\circ$ and the sine and cosine of $36^\circ$ are known exactly.

The tool is the sum and difference identity for sine:

$$\sin(A - B) = \sin A \cos B - \cos A \sin B$$

Set $A = 60^\circ$ and $B = 36^\circ$:

$$\sin 24^\circ = \sin 60^\circ \cos 36^\circ - \cos 60^\circ \sin 36^\circ$$

Substitute the known values $\sin 60^\circ = \dfrac{\sqrt{3}}{2}$, $\cos 60^\circ = \dfrac{1}{2}$, $\cos 36^\circ = \dfrac{1+\sqrt{5}}{4}$, and $\sin 36^\circ = \dfrac{\sqrt{10 - 2\sqrt{5}}}{4}$:

$$\sin 24^\circ = \frac{\sqrt{3}}{2}\cdot\frac{1+\sqrt{5}}{4} - \frac{1}{2}\cdot\frac{\sqrt{10 - 2\sqrt{5}}}{4}$$

$$\sin 24^\circ = \frac{\sqrt{3},(1+\sqrt{5}) - \sqrt{10 - 2\sqrt{5}}}{8}$$

Evaluate the radicals and this collapses to $0.4067366$, agreeing with a calculator. The reference angle here is $24^\circ$ itself, and because the terminal side lies in Quadrant I, the ASTC rule ("All" ratios positive in the first quadrant) keeps the sign positive.

Is There An Exact Value Of Sin 24 Degrees?

Yes, and this is where honesty matters more than a tidy answer. The angle $24^\circ$ is constructible: it equals $60^\circ - 36^\circ$, and both of those angles can be built with compass and straightedge (a $36^\circ$ angle comes from the regular pentagon). Because it is constructible, an exact value in radicals genuinely exists:

$$\sin 24^\circ = \frac{\sqrt{3},(1+\sqrt{5}) - \sqrt{10 - 2\sqrt{5}}}{8} = \frac{\sqrt{7 + \sqrt{5} - \sqrt{30 + 6\sqrt{5}}}}{4} \approx 0.4067$$

Both forms are correct and both equal $0.4067366$. But look at them: nested square roots stacked three deep. No student is expected to carry that around, and no exam rewards it over the decimal. This is the difference from the special angles, where $\sin 30^\circ = \tfrac{1}{2}$ is short enough to memorise.

So the takeaway is deliberately plain. An exact surd for $\sin 24^\circ$ exists, but it is impractical, and the value you write down and compute with is $0.4067$. Do not try to force $24^\circ$ into a simple fraction of roots the way $45^\circ$ or $60^\circ$ allow, because no such simple form exists.

Where Does 24 Degrees Sit On The Unit Circle?

On the unit circle, the angle $24^\circ$ is measured anticlockwise from the positive $x$-axis. The point where the terminal side meets the circle has coordinates $(\cos 24^\circ, \sin 24^\circ) = (0.9135,\ 0.4067)$. The $y$-coordinate of that point is $\sin 24^\circ$.

That is the unit-circle definition of sine: the height of the point above the horizontal axis. Since $24^\circ$ is a small positive angle in the first quadrant, the point sits just above the $x$-axis and well to the right, giving a large positive cosine ($0.9135$) and a smaller positive sine ($0.4067$).

How Do The Triangle And The Unit Circle Give The Same Value?

Sine has two faces, and $\sin 24^\circ$ should look the same from both. In a right triangle with one angle of $24^\circ$, sine is the ratio of the opposite side to the hypotenuse:

$$\sin 24^\circ = \frac{\text{opposite}}{\text{hypotenuse}}$$

Build a right triangle with hypotenuse of length $1$ and one acute angle of $24^\circ$. The side opposite that angle then has length $\sin 24^\circ = 0.4067$, and the side adjacent has length $\cos 24^\circ = 0.9135$.

Set that same triangle inside the unit circle with the hypotenuse as the radius, and the opposite side becomes the vertical height of the point, $0.4067$. Same number, two pictures. For a fuller tour of how these ratios are defined, see sin cos tan and the sine function.

What Are The Values Near Sin 24 Degrees?

Placing $24^\circ$ among its neighbours shows why $0.4067$ is the right size, and gives the related angles you can link out to. Sine climbs steadily from $0^\circ$ to $90^\circ$.

Table: Sine, cosine, and tangent for angles near 24 degrees (values to 4 dp).

Notice the mirror in the last row: $\sin 66^\circ = 0.9135 = \cos 24^\circ$, and $\cos 66^\circ = 0.4067 = \sin 24^\circ$. That is the cofunction identity at work, and the next section explains it.

Why Is Sin 24 Degrees Equal To Cos 66 Degrees?

The cofunction identity says $\sin\theta = \cos(90^\circ - \theta)$. With $\theta = 24^\circ$, the complement is $90^\circ - 24^\circ = 66^\circ$, so $\sin 24^\circ = \cos 66^\circ$. Both equal $0.4067$.

The reason is the right triangle itself.

• Two angles share one triangle. In any right triangle, the two non-right angles add to $90^\circ$. If one is $24^\circ$, the other is $66^\circ$.

• One side plays two roles. The side opposite the $24^\circ$ angle is the side adjacent to the $66^\circ$ angle. So "opposite over hypotenuse" for $24^\circ$ is the very same ratio as "adjacent over hypotenuse" for $66^\circ$.

• That is sine of one equalling cosine of the other. $\sin 24^\circ$ and $\cos 66^\circ$ are two names for one length divided by the hypotenuse.

This gives a free accuracy check. Any calculator, table, or hand computation should return the same number for $\sin 24^\circ$ and for $\cos 66^\circ$. If it does not, an error crept in.

Who Discovered The First Tables Of Sine Values?

Nobody derives $\sin 24^\circ$ from scratch each time. For two thousand years, people relied on tables, and the story of those tables is one of the great relay races in mathematics.

Two earlier figures set up the relay:

• Hipparchus of Nicaea (c. 190–120 BCE, Greece) built the first known trigonometric table, a table of chords, to predict the positions of the Sun and Moon. It was the ancestor of every sine table that followed.

• Aryabhata (476–550 CE, India) tabulated the sine (which he called jya) in steps of $3.75^\circ$, giving the world its first true sine table and the word that, mistranslated through Arabic and Latin, became "sine."

Where Is Sin 24 Degrees Used In The Real World?

Small angles like $24^\circ$ turn up wherever something tilts, oscillates, or points.

• Ramps and roofs: a slope tilted $24^\circ$ rises $\sin 24^\circ \approx 0.41$ metres for every metre of slope length, which is how builders convert an angle into a height.

• Waves and sound: alternating current, light, and sound are modelled by sine functions, and the value at a phase of $24^\circ$ (that is, $\frac{2\pi}{15}$ radians into the cycle) is $0.4067$ of the peak.

• Navigation and GPS: positions are fixed by resolving distances into horizontal and vertical parts using the sine and cosine of bearing and elevation angles.

• Computer graphics: rotating a sprite or a 3D model by $24^\circ$ multiplies its coordinates by $\sin 24^\circ$ and $\cos 24^\circ$ inside a rotation matrix.

• Astronomy: the same table-building that started with Hipparchus still underlies how the altitude of a star above the horizon is turned into a coordinate.

One number, $0.4067$, is the shared thread between a wheelchair ramp, a sound wave, and a rotating game character.

What Are The Most Common Mistakes With Sin 24 Degrees?

These four errors account for most wrong answers involving $\sin 24^\circ$, and each has a clean fix.

Reading the calculator in radian mode.

Where it slips in:

A student types "sin 24" expecting degrees, but the calculator is set to radians, and it returns $-0.9056$ instead of $0.4067$.

Don't do this:

Do not trust the display before checking the angle mode.

The correct way:

Set the calculator to degree mode (look for DEG or D on screen) when the angle is in degrees. To use radian mode instead, enter the angle as $\frac{2\pi}{15}$, not $24$. For more on the two units, see what is a radian.

Expecting a neat exact value.

Where it slips in:

A student assumes $\sin 24^\circ$ must simplify to something like $\frac{\sqrt{k}}{2}$, because $30^\circ$, $45^\circ$, and $60^\circ$ all do.

Don't do this:

Do not invent or hunt for a simple surd. The genuine exact form of $\sin 24^\circ$ is a triple-nested radical, not a special-angle value.

The correct way:

Use the decimal $0.4067$ for calculation, and only write the nested-radical form when a question explicitly asks for the exact value.

Confusing sine with cosine in the cofunction step.

Where it slips in:

A student remembers a "$90^\circ$ minus" rule but writes $\sin 24^\circ = \sin 66^\circ$, keeping the same function name.

Don't do this:

Do not leave the function unchanged. The cofunction rule swaps sine for cosine.

The correct way:

Apply $\sin\theta = \cos(90^\circ - \theta)$, so $\sin 24^\circ = \cos 66^\circ = 0.4067$, while $\sin 66^\circ = 0.9135$ is a different value.

Mishandling the sign when the angle is disguised.

Where it slips in:

A problem asks for $\sin 156^\circ$ or $\sin 204^\circ$, and a student forgets to fix the sign from the quadrant after finding the reference angle $24^\circ$.

Don't do this:

Do not copy the first-quadrant sign onto every quadrant.

The correct way:

Find the reference angle ($180^\circ - 156^\circ = 24^\circ$, or $204^\circ - 180^\circ = 24^\circ$), then set the sign by quadrant: $\sin 156^\circ = +0.4067$ (Quadrant II, sine positive), but $\sin 204^\circ = -0.4067$ (Quadrant III, sine negative).

Practice Problems On Sin 24 Degrees

Use four-decimal-place values throughout. Answers follow each problem.

1. Write $24^\circ$ in radians.
   (Answer: $24^\circ \times \frac{\pi}{180} = \frac{2\pi}{15} \approx 0.4189$ rad.)

2. Given $\sin 24^\circ = 0.4067$, state $\cos 66^\circ$.
   (Answer: $0.4067$, by the cofunction identity.)

3. A ramp is $3$ m long along its slope and tilts at $24^\circ$. How high is its top end?
   (Answer: height $= 3 \times \sin 24^\circ = 3 \times 0.4067 \approx 1.22$ m.)

4. Find $\sin 156^\circ$.
   (Answer: reference angle $180^\circ - 156^\circ = 24^\circ$; Quadrant II, so $\sin 156^\circ = +0.4067$.)

5. Using $\sin 24^\circ = 0.4067$ and $\cos 24^\circ = 0.9135$, find $\tan 24^\circ$ to 4 dp.
   (Answer: $\tan 24^\circ = \frac{0.4067}{0.9135} \approx 0.4452$.)

6. Evaluate the first three terms of the series $x - \frac{x^3}{6} + \frac{x^5}{120}$ at $x = \frac{2\pi}{15}$.
   (Answer: $\approx 0.4067$, matching $\sin 24^\circ$.)

Where Should You Go Next After Sin 24 Degrees?

Sin 24 degrees is a doorway into the wider machinery of trigonometry, and a few natural next steps open from here.

1. Sum and difference identities. The exact-value method above, $\sin(60^\circ - 36^\circ)$, generalises to any angle you can split into known parts.

2. Trigonometric ratios of specific angles. Lock in the special angles ($30^\circ$, $45^\circ$, $60^\circ$) that every derivation leans on, alongside the full trigonometric table.

3. Trigonometric ratios in radians. Get fluent in the radian form $\frac{2\pi}{15}$ that calculators and power series both need.

If your child is building these foundations, a live Bhanzu trainer teaches trigonometric values starting from the unit circle and the right triangle together, so a value like $\sin 24^\circ$ is never just a table entry, in the Bhanzu trigonometry program.