Tan 36 Degrees: Exact Value & How To Find It

What Is The Value Of Tan 36 Degrees?

Tan 36 degrees is $\sqrt{5 - 2\sqrt{5}} \approx 0.7265$. In radians the angle is $36^\circ = \frac{\pi}{5}$, so this value is also written $\tan\frac{\pi}{5}$. Both notations name the same number, and reading the angle in radians matters the moment a calculator or a computer is involved.

Here is the value in every form you are likely to need:

• Exact form: $\tan 36^\circ = \sqrt{5 - 2\sqrt{5}}$

• Decimal form: $\tan 36^\circ \approx 0.72654$ (rounded to five places)

• Radian form: $\tan\dfrac{\pi}{5} \approx 0.7265$, with $\dfrac{\pi}{5} \approx 0.6283$ rad

• Sign: positive, because $36^\circ$ lies in the first quadrant

Unlike $\tan 30^\circ$ or $\tan 45^\circ$, this one is rarely memorised. What makes it worth knowing is that it has an exact surd at all, and the reason traces straight back to a shape most children can draw.

How Do You Find Tan 36 Degrees?

There are two honest ways to reach the value: read it off the unit circle, or build it from the right-triangle ratio. A good trigonometry student should recognise both, because the same number should mean the same thing whether it comes from a circle or a triangle.

From the right triangle. In a right triangle containing a $36^\circ$ angle, the tangent is the side opposite the angle divided by the side next to it (adjacent).

$$\tan 36^\circ = \frac{\text{opposite}}{\text{adjacent}}$$

Measure a right triangle with a $36^\circ$ corner and the opposite side comes out about $0.7265$ times the adjacent side, every time, whatever the triangle's overall size. That constancy is the whole idea of a trigonometric ratio.

From the unit circle. On a circle of radius $1$, the point at $36^\circ$ has coordinates $(\cos 36^\circ, \sin 36^\circ)$, and the tangent is the $y$-coordinate divided by the $x$-coordinate.

$$\tan 36^\circ = \frac{\sin 36^\circ}{\cos 36^\circ} = \frac{0.5878}{0.8090} \approx 0.7265$$

Both routes land on the same $0.7265$, which is exactly what the tangent function promises: one ratio, two pictures.

Where Does 36° Sit On The Unit Circle?

The angle $36^\circ$ is measured anticlockwise from the positive $x$-axis, so it sits in the first quadrant, between $0^\circ$ and $90^\circ$. In the first quadrant every basic ratio is positive, which is why $\tan 36^\circ$ carries a plus sign with no extra work.

The point where the $36^\circ$ line meets the circle is $(0.8090,\ 0.5878)$. Reading the tangent as slope, you rise $0.5878$ for every $0.8090$ you run, and that slope is $0.7265$.

How Do You Derive The Exact Value Of Tan 36 Degrees?

The exact surd comes from the two pentagon values that trigonometry gives for $36^\circ$:

$$\sin 36^\circ = \frac{\sqrt{10 - 2\sqrt{5}}}{4}, \qquad \cos 36^\circ = \frac{1 + \sqrt{5}}{4}$$

That cosine is the golden ratio in disguise, since $\frac{1+\sqrt5}{2} = \varphi$ and $\cos 36^\circ = \frac{\varphi}{2}$. Dividing sine by cosine and then clearing the roots produces the clean answer.

Start from the ratio:

$$\tan 36^\circ = \frac{\sin 36^\circ}{\cos 36^\circ} = \frac{\sqrt{10 - 2\sqrt{5}}}{1 + \sqrt{5}}$$

Square both sides to remove the outer root:

$$\tan^2 36^\circ = \frac{10 - 2\sqrt{5}}{(1 + \sqrt{5})^2} = \frac{10 - 2\sqrt{5}}{6 + 2\sqrt{5}}$$

Rationalise by multiplying the top and bottom by $6 - 2\sqrt{5}$:

$$\tan^2 36^\circ = \frac{(10 - 2\sqrt{5})(6 - 2\sqrt{5})}{(6 + 2\sqrt{5})(6 - 2\sqrt{5})} = \frac{80 - 32\sqrt{5}}{16} = 5 - 2\sqrt{5}$$

Take the positive root, since $36^\circ$ is in the first quadrant:

$$\tan 36^\circ = \sqrt{5 - 2\sqrt{5}} \approx 0.7265$$

A quick sanity check: $2\sqrt{5} = 4.4721$, so $5 - 2\sqrt{5} = 0.5279$, and $\sqrt{0.5279} = 0.7265$. The exact form and the decimal agree.

What Are The Tan Values Around 36°?

The pentagon family ($18^\circ$, $36^\circ$, $54^\circ$, $72^\circ$) all carry exact surds, and they mirror each other. Compare them against the trigonometric table of the more familiar angles.

Table: The pentagon-family angles, with exact and decimal values.

Notice how the $36^\circ$ and $54^\circ$ rows swap their sine and cosine. That swap is the cofunction relationship, and it means $\tan 36^\circ = \cot 54^\circ$, since $36^\circ$ and $54^\circ$ add to $90^\circ$.

Why Is Tan 36 Degrees Equal To That Surd?

The short answer: $36^\circ$ is the angle a regular pentagon hands you, and the pentagon is built on the golden ratio, which is itself a square-root expression. When the geometry hides a square root, so does the trigonometry.

• A regular pentagon has a central angle of $\frac{360^\circ}{5} = 72^\circ$, and $36^\circ$ is exactly half of that, the angle at the tip of each point of a five-pointed star.

• Splitting the pentagon into golden triangles forces the ratio $\frac{1+\sqrt5}{2}$ to appear, which is why $\cos 36^\circ = \frac{1+\sqrt5}{4}$ carries a $\sqrt5$.

• An angle whose exact value can be written with whole numbers, arithmetic, and square roots is called constructible, meaning you could draw it with only a compass and straightedge. The pentagon is constructible, so $36^\circ$ is too.

This is the real difference between $36^\circ$ and an angle like $37^\circ$. Thirty-seven degrees has no neat closed form, only a decimal. Thirty-six degrees inherits a clean surd from a shape you can construct, and that is why it earns an exact answer.

Who Discovered The Value Of Tan 36 Degrees?

Long before anyone wrote "tan," astronomers needed the length of the chord that a $36^\circ$ arc cuts across a circle. Working that out for the pentagon and its cousin the decagon was one of the oldest hard problems in trigonometry.

Two other figures shaped this same value:

• Hipparchus of Nicaea (c. 190–120 BCE, Greece) is often called the father of trigonometry, and his lost chord tables were the first known attempt to turn angles into numbers.

• Aryabhata (476–550 CE, India) compiled an early table of half-chords, the direct ancestor of the sine (jya), giving trigonometry the function-based form we still use.

Where Is Tan 36 Degrees Used In The Real World?

The pentagon angle shows up wherever five-fold symmetry does, which is more places than most people expect.

• Computer graphics and games: drawing a perfect five-pointed star, a common icon and rating symbol, means turning by multiples of $36^\circ$ and using its tangent to place each point.

• Architecture and design: pentagonal floor plans, domes, and tiling patterns rely on $36^\circ$ and $72^\circ$ angles to close cleanly.

• Crystallography and materials: quasicrystals and structures with five-fold symmetry (unknown in ordinary crystals) are described using pentagon angles.

• Engineering: five-spoke wheels, gears, and bolt circles space their parts $72^\circ$ apart, and laying them out uses the tangent of the half-angle, $36^\circ$.

One angle, borrowed from a starfish and a paper star, quietly sets the geometry of screens, buildings, and machines. That reach is why an "obscure" value is worth an exact form.

What Are The Most Common Mistakes With Tan 36 Degrees?

These four errors account for most wrong answers involving $36^\circ$, and each has a clean fix.

Leaving the calculator in the wrong angle mode.

Where it slips in:

A student types tan(36) while the calculator or a spreadsheet is set to radians, and reads off $\approx 7.75$ instead of $0.7265$.

Don't do this:

Do not trust the number before checking the mode. $\tan(36 \text{ rad})$ and $\tan(36^\circ)$ are completely different values.

The correct way:

Set the device to degree mode for $36^\circ$, or convert first: $36^\circ = \frac{\pi}{5}$ rad, then evaluate $\tan\frac{\pi}{5}$. See trigonometric ratios in radians for the conversion.

Getting the sign wrong for a related angle.

Where it slips in:

A student assumes $\tan 144^\circ$ or $\tan 216^\circ$ has the same sign as $\tan 36^\circ$ because the reference angle is $36^\circ$.

Don't do this:

Do not copy the first-quadrant sign into other quadrants. Tangent is positive in the first and third quadrants and negative in the second and fourth (the ASTC rule).

The correct way:

Find the reference angle, then apply ASTC. $\tan 144^\circ = -\tan 36^\circ = -0.7265$ (second quadrant), while $\tan 216^\circ = +\tan 36^\circ = 0.7265$ (third quadrant).

Misidentifying the reference angle.

Where it slips in:

For an angle such as $216^\circ$, a student subtracts from the wrong benchmark and uses $54^\circ$ instead of $36^\circ$.

Don't do this:

Do not guess the reference angle. It is the acute angle between the terminal side and the $x$-axis, not the $y$-axis.

The correct way:

In the third quadrant the reference angle is (angle $- 180^\circ$). For $216^\circ$ that is $216^\circ - 180^\circ = 36^\circ$, so the tangent matches $\tan 36^\circ$ in size.

Confusing the cofunction with the same-name function.

Where it slips in:

A student writes $\tan 36^\circ = \tan 54^\circ$, mixing up the cofunction rule.

Don't do this:

Do not pair tangent with tangent across $90^\circ$. The cofunction of tangent is cotangent.

The correct way:

Because $36^\circ + 54^\circ = 90^\circ$, the rule gives $\tan 36^\circ = \cot 54^\circ \approx 0.7265$, while $\tan 54^\circ \approx 1.3764$ is a different number. This follows from the trigonometric ratios of complementary angles.

Practice Problems On Tan 36 Degrees

Work each one, then check against the answer.

1. Convert $36^\circ$ to radians.
   (Answer: $\frac{\pi}{5} \approx 0.6283$ rad.)

2. Using $\tan 36^\circ = \sqrt{5 - 2\sqrt5}$, state $\cot 36^\circ$ as a decimal.
   (Answer: $\cot 36^\circ = \frac{1}{\tan 36^\circ} \approx 1.3764$.)

3. Use the cofunction rule to write $\cot 54^\circ$.
   (Answer: $\cot 54^\circ = \tan 36^\circ \approx 0.7265$.)

4. Evaluate $\tan 216^\circ$.
   (Answer: reference angle $36^\circ$, third quadrant, so $\tan 216^\circ = +0.7265$.)

5. Evaluate $\tan 144^\circ$.
   (Answer: reference angle $36^\circ$, second quadrant, so $\tan 144^\circ = -0.7265$.)

6. A right triangle has a $36^\circ$ angle whose adjacent side is $10$ cm. Find the opposite side.
   (Answer: opposite $= 10 \tan 36^\circ \approx 7.265$ cm.)

Where Should You Go Next After Tan 36 Degrees?

The value of $36^\circ$ opens onto the wider system of angle values, and a few natural doors lead outward.

1. Trigonometric ratios of specific angles. See how $30^\circ$, $45^\circ$, and $60^\circ$ get their exact values, and where $36^\circ$ fits alongside them.

2. Unit circle with tangent. Watch the tangent grow and change sign as the angle sweeps around the full circle.

3. Sin cos tan. Revisit the three core ratios and how they connect on the triangle and the circle.

If your child is building these foundations, a live Bhanzu trainer teaches special angles starting from the "why" (the pentagon and the circle the value was born from) in the Bhanzu trigonometry program.