Sin 72 Degrees: Exact Value & Unit Circle

What Is The Value Of Sin 72 Degrees?

Sin 72 degrees is $\sin 72^\circ = \dfrac{\sqrt{10 + 2\sqrt{5}}}{4}$, which is approximately $0.9511$ to four decimal places. The angle in radians is $72^\circ = \dfrac{2\pi}{5} \approx 1.2566$ radians, and because $72^\circ$ lies in the first quadrant, the value is positive.

Two things make this angle worth a full article:

• It has an exact surd. Most angles have only a decimal. The angle $72^\circ$ is constructible, meaning you can build it with a compass and straightedge, so its sine is an exact combination of square roots.

• It is the pentagon angle. A full turn split into five equal parts gives $360^\circ \div 5 = 72^\circ$. That single fact links $\sin 72^\circ$ to the regular pentagon and the golden ratio.

You can also read the value from a right triangle and from the unit circle, the two anchors used throughout this article. In a right triangle, $\sin 72^\circ$ is the ratio of the side opposite a $72^\circ$ angle to the hypotenuse. On the unit circle, it is the height (the $y$-coordinate) of the point reached after turning $72^\circ$ from the positive $x$-axis.

How Do You Find Sin 72 Degrees?

The clean way to reach the exact value starts one step below, at $18^\circ$, then climbs up with a co-function identity. The idea: $18^\circ$ fits five times into $90^\circ$, which turns the angle into a solvable equation.

Let $\theta = 18^\circ$. Then $5\theta = 90^\circ$, so $2\theta = 90^\circ - 3\theta$. Taking sine of both sides and using $\sin(90^\circ - x) = \cos x$:

$$\sin 2\theta = \cos 3\theta$$

Expand each side with the double-angle and triple-angle formulas:

$$2\sin\theta\cos\theta = 4\cos^{3}\theta - 3\cos\theta$$

Divide through by $\cos\theta$ (it is not zero here), then replace $\cos^{2}\theta$ with $1 - \sin^{2}\theta$:

$$2\sin\theta = 4(1 - \sin^{2}\theta) - 3$$

$$4\sin^{2}\theta + 2\sin\theta - 1 = 0$$

Solve this quadratic in $\sin\theta$ with the quadratic formula, keeping the positive root:

$$\sin 18^\circ = \frac{-2 + \sqrt{4 + 16}}{8} = \frac{\sqrt{5} - 1}{4}$$

Now use the co-function identity $\sin 72^\circ = \cos 18^\circ$, and find $\cos 18^\circ$ from $\cos 18^\circ = \sqrt{1 - \sin^{2}18^\circ}$:

$$\sin 72^\circ = \cos 18^\circ = \sqrt{1 - \left(\tfrac{\sqrt5 - 1}{4}\right)^{2}} = \frac{\sqrt{10 + 2\sqrt{5}}}{4}$$

Final answer: $\sin 72^\circ = \dfrac{\sqrt{10 + 2\sqrt{5}}}{4} \approx 0.9511$.

To confirm the sign without any algebra, use the ASTC (or CAST) rule for quadrants. The angle $72^\circ$ sits in Quadrant I, where all ratios are positive, so $\sin 72^\circ$ comes out positive. Its reference angle is $72^\circ$ itself, since first-quadrant angles are their own reference angle.

Where Does 72 Degrees Sit On The Unit Circle?

On the unit circle, an angle is measured anticlockwise from the positive $x$-axis, and the sine of that angle is the height of the point where the terminal arm meets the circle, its $y$-coordinate. Turning $72^\circ$ lands you high and slightly right of centre, at the point $(\cos 72^\circ,\ \sin 72^\circ) = (0.3090,\ 0.9511)$.

That height, $0.9511$, is $\sin 72^\circ$. The point is close to the top of the circle because $72^\circ$ is close to $90^\circ$, where the sine reaches its maximum of $1$.

The same value shows up as a right-triangle ratio. Drop the vertical from the point $P$ to the $x$-axis and you form a right triangle with hypotenuse $1$ (the radius). The side opposite the $72^\circ$ angle has length $0.9511$, so $\sin 72^\circ = \dfrac{\text{opposite}}{\text{hypotenuse}} = \dfrac{0.9511}{1}$. Triangle and circle give the same number, which is the whole point of tying the two pictures together.

What Is Sin 72 Degrees In Terms Of Other Ratios?

Because $72^\circ$ belongs to the pentagon family (the multiples of $18^\circ$), its ratios connect neatly to the angles around it. A few worth knowing:

• Co-function: $\sin 72^\circ = \cos 18^\circ$, since $18^\circ$ and $72^\circ$ are complementary angles that add to $90^\circ$.

• Cosine of the same angle: $\cos 72^\circ = \dfrac{\sqrt{5} - 1}{4} \approx 0.3090$.

• Tangent: $\tan 72^\circ = \dfrac{\sin 72^\circ}{\cos 72^\circ} = \sqrt{5 + 2\sqrt{5}} \approx 3.0777$.

Here is the full pentagon family in one place, with each angle in degrees and radians, exact where a clean surd exists.

Table: The pentagon-family angles (multiples of 18 degrees) with exact and decimal sine and cosine.

Notice the mirror: $\sin 18^\circ = \cos 72^\circ$ and $\sin 72^\circ = \cos 18^\circ$. That is the co-function rule at work across the table. For the standard reference set, see the trigonometric table and the trigonometric ratios of specific angles.

Why Is Sin 72 Degrees Equal To That Surd?

The surd is not an accident of the algebra. It traces straight back to a regular pentagon, a shape you can draw.

• The pentagon sets the angle. A regular pentagon splits a full circle into five equal central angles of $72^\circ$ each. Any exact fact about the pentagon becomes an exact fact about $72^\circ$, and the pentagon is buildable with compass and straightedge.

• The golden ratio sets the numbers. The diagonals of a regular pentagon cut each other in the golden ratio $\varphi = \dfrac{1 + \sqrt{5}}{2}$. That is where the $\sqrt5$ inside $\sin 72^\circ$ comes from. In fact $\cos 36^\circ = \dfrac{\varphi}{2}$, and $\sin 72^\circ$ is built from the same family of lengths.

• Constructible means exact. Because the pentagon can be constructed, every angle it produces has a value expressible in whole numbers, arithmetic, and square roots. That is exactly what a surd is, so $\sin 72^\circ$ has a closed form while an angle like $\sin 73^\circ$ does not.

The short version: $\sin 72^\circ$ is clean because $72^\circ$ is the pentagon's own angle, and the pentagon is one of the shapes the Greeks could draw with the two oldest tools in geometry. The angles in a pentagon carry the golden ratio, and the golden ratio carries the $\sqrt5$.

Who Discovered The Value Of Sin 72 Degrees?

Nobody woke up one morning and computed $\sin 72^\circ$ as a decimal. The value arrived slowly, first as the length of a chord in a circle, long before the word "sine" existed.

Two later figures carried the same work forward:

• Claudius Ptolemy (c. 100 – c. 170 CE, Roman Egypt) refined the chord table in his Almagest, computing chords in half-degree steps and explicitly using the regular pentagon and decagon to pin down the $36^\circ$ and $72^\circ$ values.

• Aryabhata (476 – 550 CE, India) replaced chords with the half-chord, the quantity we now call sine, and tabulated it in his Aryabhatiya, giving the function the form and the name (jya) that eventually became "sine."

Where Is Sin 72 Degrees Used In The Real World?

The $72^\circ$ turn is everywhere five-fold symmetry appears, and its sine is what turns that turn into a measurable height or width.

• Design and architecture: five-pointed stars, pentagon tiling, and the Pentagon building itself all rest on the $72^\circ$ central angle, and laying them out on a grid needs $\sin 72^\circ$ for the coordinates.

• Nature and phyllotaxis: many flowers set petals and seeds around a turn related to the golden angle, and the pentagon's $72^\circ$ appears in the five-petal blooms and five-fold fruit cores you can slice open and check.

• Signal processing: a five-point rotation is one step of a five-term Fourier calculation, where $\sin 72^\circ$ is a fixed coefficient used to break a wave into its frequencies.

• Computer graphics: drawing or rotating any regular pentagon or star polygon on screen places each vertex using the sine and cosine of multiples of $72^\circ$.

• Engineering: five-bladed fans, propellers, and gear layouts space their parts $72^\circ$ apart, and the stress and balance calculations use the sine of that spacing.

One angle, born from a shape the Greeks could draw, still positions petals, pixels, and propeller blades. That reach across fields is what makes a single trigonometric value worth knowing exactly.

What Are The Most Common Mistakes With Sin 72 Degrees?

These four slips account for most wrong answers on this angle. Each is easy to avoid once you have seen it.

Leaving the calculator in radian mode.

Where it slips in:

A student types sin(72) expecting $0.9511$ but the calculator is set to radians, and it returns about $0.2538$ instead.

Don't do this:

Do not trust the number until you have checked the angle unit. $72$ radians is a completely different angle from $72^\circ$.

The correct way:

Set the mode to degrees for $\sin 72^\circ$, or convert first: $72^\circ = \dfrac{2\pi}{5}$ radians, then evaluate $\sin\left(\dfrac{2\pi}{5}\right)$. Either way you get $0.9511$. If you are unsure what a radian is, review what is a radian.

Thinking sin 72 degrees is negative.

Where it slips in:

A student assumes any "large" angle must be past the halfway point and give a negative sine.

Don't do this:

Do not guess the sign. $72^\circ$ is still short of $90^\circ$, so it stays in Quadrant I.

The correct way:

Use ASTC. In Quadrant I all ratios are positive, so $\sin 72^\circ = +0.9511$. Sine only turns negative below the $x$-axis, from $180^\circ$ to $360^\circ$.

Confusing sin 72 degrees with sin 36 degrees.

Where it slips in:

Both belong to the pentagon family and both surds contain $\sqrt5$, so their values get swapped.

Don't do this:

Do not treat $\sqrt{10 + 2\sqrt5}$ and $\sqrt{10 - 2\sqrt5}$ as interchangeable. The sign in the middle changes the value.

The correct way:

Match the size to the angle. The larger angle has the larger sine: $\sin 72^\circ = \dfrac{\sqrt{10 + 2\sqrt5}}{4} \approx 0.9511$, while $\sin 36^\circ = \dfrac{\sqrt{10 - 2\sqrt5}}{4} \approx 0.5878$.

Reading the co-function backwards.

Where it slips in:

A student remembers a co-function link but writes $\sin 72^\circ = \cos 72^\circ$ instead of $\cos 18^\circ$.

Don't do this:

Do not pair an angle with its own cosine. The co-function uses the complement, the angle that completes $90^\circ$.

The correct way:

Subtract from $90^\circ$ first: $\sin 72^\circ = \cos(90^\circ - 72^\circ) = \cos 18^\circ$. The two complementary angles $18^\circ$ and $72^\circ$ swap sine and cosine.

Practice Problems On Sin 72 Degrees

Work each one, then check against the answer. Use $\sin 72^\circ = \dfrac{\sqrt{10 + 2\sqrt5}}{4} \approx 0.9511$ where needed.

1. Write $72^\circ$ in radians.
   (Answer: $\dfrac{2\pi}{5} \approx 1.2566$ radians.)

2. State $\cos 18^\circ$ using a co-function identity.
   (Answer: $\cos 18^\circ = \sin 72^\circ = \dfrac{\sqrt{10 + 2\sqrt5}}{4} \approx 0.9511$.)

3. A right triangle has a $72^\circ$ angle and hypotenuse $10$ cm. Find the side opposite the $72^\circ$ angle.
   (Answer: $10 \times \sin 72^\circ \approx 9.511$ cm.)

4. Evaluate $\tan 72^\circ$ using $\sin 72^\circ$ and $\cos 72^\circ = 0.3090$.
   (Answer: $\dfrac{0.9511}{0.3090} \approx 3.0777$.)

5. Which is larger, $\sin 72^\circ$ or $\sin 78^\circ$?
   (Answer: $\sin 78^\circ \approx 0.9781$ is larger, since sine increases from $0^\circ$ to $90^\circ$.)

6. Find $\sin 72^\circ + \cos 72^\circ$ as a decimal.
   (Answer: $0.9511 + 0.3090 = 1.2601$.)

Where Should You Go Next After Sin 72 Degrees?

The pentagon angle opens several natural doors, each backed by a reference page.

1. Cofunction identities. The rule that turned $\sin 72^\circ$ into $\cos 18^\circ$, with every sine-cosine pair that completes $90^\circ$.

2. Unit circle with tangent. See where $72^\circ$ and every other angle land, and how sine, cosine, and tangent read off the circle.

3. Trigonometric ratios of specific angles. The full set of exact values, from the common $30^\circ$-$45^\circ$-$60^\circ$ angles out to the pentagon family.

If your child is building these values into real fluency, a live Bhanzu trainer teaches them from the shape up, starting with why the pentagon makes $72^\circ$ special, in the Bhanzu math program.