Tan 40 Degrees: Value, Radians and How To Find It

What Is The Value Of Tan 40 Degrees?

The value of Tan 40 Degrees is approximately $0.8391$, correct to four decimal places (a longer decimal is $0.83909963$). In a right triangle, the tangent of an angle is the length of the side opposite the angle divided by the side adjacent to it, so $\tan 40^\circ$ is the ratio you get when one acute angle measures $40^\circ$.

The angle can be written two ways, and instructional work should always show both:

$$\tan 40^\circ = \tan\left(\frac{2\pi}{9}\right) \approx 0.8391$$

Here $40^\circ$ in radians is $40 \times \frac{\pi}{180} = \frac{2\pi}{9} \approx 0.6981$ radians. The value is positive because $40^\circ$ is a first-quadrant angle. For the ratio itself and how it links to sine and cosine, see sin cos tan and the fuller tangent function page.

How Do You Find Tan 40 Degrees?

Since $40^\circ$ is not one of the standard angles, you find it from its two building blocks, sine and cosine, because tangent is their ratio.

$$\tan 40^\circ = \frac{\sin 40^\circ}{\cos 40^\circ} = \frac{0.6428}{0.7660} \approx 0.8391$$

Two steps fix the sign and the size:

• Reference angle. The reference angle is the acute angle to the nearest part of the x-axis. For $40^\circ$ that reference angle is $40^\circ$ itself, so no adjustment is needed.

• Quadrant sign. Use the mnemonic All Students Take Calculus (the CAST or ASTC rule): in the first quadrant All ratios are positive. So $\tan 40^\circ$ is positive.

From a right triangle, build a triangle with one angle of $40^\circ$, measure the opposite and adjacent sides, and divide. From the trigonometric ratios, $\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$, which gives the same $0.8391$. You can read the value straight from the trigonometric table as well.

Where Does 40 Degrees Sit On The Unit Circle?

On the unit circle (radius $1$), an angle of $40^\circ$ measured from the positive x-axis lands on a point whose coordinates are $(\cos 40^\circ, \sin 40^\circ) \approx (0.7660, 0.6428)$. The tangent is the y-coordinate divided by the x-coordinate.

$$\tan 40^\circ = \frac{y}{x} = \frac{\sin 40^\circ}{\cos 40^\circ} = \frac{0.6428}{0.7660} \approx 0.8391$$

Because both coordinates are positive in the first quadrant, their ratio is positive, which confirms the sign from the CAST rule. For the geometric picture of tangent as a length on the circle, see unit circle with tangent, and for the meaning of the radian measure, see what is a radian.

Can Tan 40 Degrees Be Written As An Exact Value?

There is no simple surd form for $\tan 40^\circ$, and being honest about that matters. The angles with clean root values ($30^\circ$, $45^\circ$, $60^\circ$) come from triangles you can construct with a compass and straightedge. The angle $40^\circ$ is one ninth of $360^\circ$, and a regular nine-sided figure cannot be drawn that way, so $\tan 40^\circ$ is non-constructible.

There is still an exact relationship, just not a tidy one. The triple-angle identity ties $40^\circ$ to $120^\circ$, because $3 \times 40^\circ = 120^\circ$ and $\tan 120^\circ = -\sqrt{3}$. Writing $t = \tan 40^\circ$:

$$\tan 3\theta = \frac{3t - t^{3}}{1 - 3t^{2}}$$

$$\frac{3t - t^{3}}{1 - 3t^{2}} = -\sqrt{3}$$

$$t^{3} - 3\sqrt{3},t^{2} - 3t + \sqrt{3} = 0$$

That cubic is what $\tan 40^\circ$ truly satisfies, and it is the kind of cubic whose real roots cannot be reduced to a finite chain of square roots. So the correct exact answer to record is the decimal $0.8391$, not an invented radical.

Two exact facts are still worth keeping. First, the cofunction identity: since $40^\circ$ and $50^\circ$ add to $90^\circ$,

$$\tan 40^\circ = \tan(90^\circ - 50^\circ) = \cot 50^\circ \approx 0.8391$$

Second, a calculator or a printed table does not use a surd at all. It sums a fast-converging power series for sine and cosine (a handful of terms is enough for four-place accuracy), or runs a shift-and-add routine, then divides the two results. This is the same machinery behind cofunction identities and trigonometric ratios of complementary angles.

Table: The tangent of 40° compared with nearby angles, in degrees and radians.

The neighbours with tidy forms are tan 30 degrees, tan 45 degrees, and tan 60 degrees. Tan 50 degrees, like $\tan 40^\circ$, has no simple surd.

Why Is Tan 40 Degrees Positive?

The sign of any tangent is decided by the quadrant of the angle, and $40^\circ$ lands squarely in the first.

• First-quadrant angle. $40^\circ$ is between $0^\circ$ and $90^\circ$, so its terminal side is in Quadrant I.

• Both coordinates positive. On the unit circle the point is $(0.7660, 0.6428)$, and a positive $y$ over a positive $x$ gives a positive ratio.

• Slope reading. Tangent measures the steepness of the radius, and a line rising to the right through the first quadrant has positive slope.

The value $0.8391$ being less than $1$ also tells you something: at $40^\circ$ the opposite side is shorter than the adjacent side. Tangent only reaches $1$ at $45^\circ$, where the two sides are equal, and grows without bound as the angle nears $90^\circ$.

Who Discovered The Tangent Ratio?

The tangent began not as a circle idea but as a shadow. Ancient astronomers noticed that the length of a shadow cast by a fixed pole changes with the sun's angle, and that shadow-to-height ratio is exactly what we now call the tangent.

Two more figures shaped the ratio behind $\tan 40^\circ$:

• Aryabhata (476–550 CE, India) compiled an influential table of sine values (which he called jya), carrying the subject forward from chords toward the ratios used now.

• Muhammad ibn Musa al-Khwarizmi and later Islamic astronomers refined shadow tables, and the Latin word tangent (from "touching") entered European mathematics in the sixteenth century.

Where Is Tan 40 Degrees Used In The Real World?

A tangent of about $0.84$ shows up wherever a slope, a height, or a line of sight sits near a $40^\circ$ angle.

• Ramps and roofs. A roof pitched at $40^\circ$ rises $0.84$ units for every $1$ unit of horizontal run, which builders use to size rafters and gutters.

• Sightlines and safety. Surveyors and drone pilots use $\tan 40^\circ$ to convert a viewing angle into a height, or a height into a ground distance.

• Engineering and mechanics. The angle of a conveyor, a loading ramp, or a support strut is chosen from the tangent so the slope stays within a safe limit.

• Cameras and graphics. A camera's field of view is set with tangents of half-angles, and $40^\circ$ is a common angle for a natural-looking lens.

• Navigation and astronomy. Elevation angles of stars and landmarks turn into distances through the same shadow-length idea Hipparchus started with.

One ratio connects rooftops, drones, camera lenses, and the night sky, which is what makes trigonometry worth learning once and using everywhere.

What Are The Most Common Mistakes With Tan 40 Degrees?

These four slips account for most wrong answers on angle values like $40^\circ$.

Leaving the calculator in radian mode.

Where it slips in:

A student types tan(40) while the calculator is set to radians and reads off $1.1578$, then writes it down as $\tan 40^\circ$.

Don't do this:

Do not trust the display before checking the angle unit. In radians, $40$ is a completely different angle from $40^\circ$.

The correct way:

Set the calculator to degree mode for $\tan 40^\circ$, or convert first: $40^\circ = \frac{2\pi}{9}$ radians, and then $\tan\left(\frac{2\pi}{9}\right) \approx 0.8391$.

Hunting for a surd that does not exist.

Where it slips in:

A student assumes every angle has a root form like $30^\circ$ or $60^\circ$ and spends time trying to write $\tan 40^\circ$ as a fraction with square roots.

Don't do this:

Do not invent a radical. The angle $40^\circ$ is non-constructible, so no finite square-root expression equals its tangent.

The correct way:

Record the decimal $0.8391$ (or $\cot 50^\circ$) as the exact answer, and reserve surd forms for the special angles that truly have them.

Confusing cot 50° with tan 50°.

Where it slips in:

A student remembers a "50" is involved and writes $\tan 40^\circ = \tan 50^\circ$, mixing up the cofunction.

Don't do this:

Do not swap the function name. $\tan 50^\circ \approx 1.1918$, which is not $0.8391$.

The correct way:

Use the cofunction rule exactly: $\tan 40^\circ = \cot 50^\circ$, because $40^\circ$ and $50^\circ$ are complementary.

Misplacing the reference angle for related angles.

Where it slips in:

Asked for $\tan 140^\circ$ or $\tan 220^\circ$, a student reuses $0.8391$ with the wrong sign because the quadrant was not checked.

Don't do this:

Do not carry the first-quadrant sign into other quadrants without the CAST rule.

The correct way:

Find the reference angle ($40^\circ$ in these cases), take $\tan 40^\circ = 0.8391$, then apply the quadrant sign, so $\tan 140^\circ = -0.8391$ and $\tan 220^\circ = +0.8391$.

Practice Problems On Tan 40 Degrees

Give answers to four decimal places where a decimal is needed.

1. Write $40^\circ$ in radians.
   (Answer: $\frac{2\pi}{9} \approx 0.6981$ radians.)

2. Use $\sin 40^\circ = 0.6428$ and $\cos 40^\circ = 0.7660$ to find $\tan 40^\circ$.
   (Answer: $\frac{0.6428}{0.7660} \approx 0.8391$.)

3. State $\tan 40^\circ$ as a cofunction of a complementary angle.
   (Answer: $\cot 50^\circ$.)

4. Find $\tan 140^\circ$.
   (Answer: reference angle $40^\circ$, Quadrant II, so $-0.8391$.)

5. A ramp rises at $40^\circ$. If it covers a horizontal distance of $5$ m, how high is its top?
   (Answer: $5 \times \tan 40^\circ \approx 4.20$ m.)

6. True or false: $\tan 40^\circ$ can be written as a simple surd.
   (Answer: False, $40^\circ$ is non-constructible.)

Where Should You Go Next After Tan 40 Degrees?

A single value opens onto the rest of trigonometry, and a few natural doors lead outward.

1. Sin cos tan. See how tangent is built from sine and cosine, the ratio behind every value on this page.

2. Trigonometric table. Look up tangent, sine, and cosine for the standard angles in one place, then compare them with $40^\circ$.

3. Cofunction identities. Learn why complementary angles trade sine for cosine and tangent for cotangent, the rule that gives $\tan 40^\circ = \cot 50^\circ$.

If your child is building these foundations, a live Bhanzu trainer teaches trigonometry from the ratios up, starting with why the tangent works before any value is memorised, in the Bhanzu trigonometry program.