Tan 0 Degrees : Value, Proof, and Why It Equals 0

What Does Tan 0 Degrees Mean?

Tangent is one of the three core trigonometric ratios. In a right triangle, the tangent of an angle is the side opposite the angle divided by the side adjacent to it. So $\tan 0^\circ$ asks: in a right triangle with a $0^\circ$ angle, how tall is the opposite side compared with the adjacent one?

At $0^\circ$ the triangle flattens completely, the opposite side has length $0$, and the ratio collapses to $\dfrac{0}{\text{adjacent}} = 0$. On the unit circle, a circle of radius $1$ centred at the origin, tangent is the $y$-coordinate divided by the $x$-coordinate of the point where the angle's radius meets the circle. At $0^\circ$ that point is $(1, 0)$, so tangent is $\dfrac{0}{1} = 0$.

Where Does Tan 0 Degrees Show Up?

Tangent of an angle measures slope, so $\tan 0^\circ = 0$ is the slope of anything perfectly level: a flat road, a still water surface, a table top. A ramp built at $0^\circ$ has a gradient of $\tan 0^\circ = 0$, which is another way of saying it does not rise at all over its run.

The value also marks where the tangent curve crosses the origin. Plot $y = \tan x$ and the graph passes through $(0, 0)$, climbing away on either side. That crossing is the reference point every tangent formula is measured against, and it sits on the same unit circle used for every other special angle.

Standard-Angle Reference Table

Zero degrees is the anchor of the tangent table, and reading down from it shows how fast tangent climbs. Here are the standard first-quadrant angles in degrees and radians.

Tangent starts at $0$ and grows without bound as the angle approaches $90^\circ$, where the ratio has a zero on the bottom and stops being defined. A full trigonometric ratio table lists sine, cosine, and tangent for each of these angles side by side.

How Do You Find The Exact Value Of Tan 0 Degrees?

There are two clean routes, and both land on $0$.

Method 1: The sine-over-cosine ratio.

Tangent is defined as sine divided by cosine:

$$\tan\theta = \frac{\sin\theta}{\cos\theta}$$

At $0^\circ$, $\sin 0^\circ = 0$ and $\cos 0^\circ = 1$, so:

$$\tan 0^\circ = \frac{\sin 0^\circ}{\cos 0^\circ} = \frac{0}{1} = 0$$

Method 2: The unit circle.

Set the radius to $1$ and leave it lying flat along the positive $x$-axis, which is the $0^\circ$ position. The tip sits at $(1, 0)$.

$$\tan 0^\circ = \frac{y\text{-coordinate}}{x\text{-coordinate}} = \frac{0}{1} = 0$$

The two methods agree because the unit circle's $y$-coordinate is $\sin\theta$ and its $x$-coordinate is $\cos\theta$, so $\dfrac{y}{x}$ is exactly $\dfrac{\sin\theta}{\cos\theta}$. Because tangent repeats every $180^\circ$, the same $0$ shows up again at $180^\circ$ and at $\tan 2\pi$. See tan 2π and tan π, which are both $0$ for this reason.

Examples Of Tan 0 Degrees

Example 1

Evaluate $5\tan 0^\circ + 3$.

$$5\tan 0^\circ + 3 = 5 \times 0 + 3 = 3$$

Example 2

Find the value of $\dfrac{\tan 0^\circ}{\cos 0^\circ}$.

Wrong attempt. A student sees a fraction with a trig value on top and a trig value on the bottom and assumes it must be undefined, the way $\tan 90^\circ$ is.

That reasoning breaks when you check the bottom. A ratio is only undefined when its denominator is $0$. Here the denominator is $\cos 0^\circ = 1$, which is fine.

Correct. Substitute both values:

$$\frac{\tan 0^\circ}{\cos 0^\circ} = \frac{0}{1} = 0$$

The result is $0$, cleanly defined. Tangent is undefined only where cosine is $0$ (at $90^\circ$), not where sine is $0$.

Example 3

A straight path rises at an angle of $0^\circ$ over a horizontal run of $12$ m. How much does it climb?

The rise equals run $\times \tan(\text{angle})$:

$$\text{rise} = 12 \times \tan 0^\circ = 12 \times 0 = 0 \text{ m}$$

A $0^\circ$ path is level, so it climbs $0$ metres.

Example 4

Verify that $\tan 0^\circ = \sin 0^\circ \times \sec 0^\circ$.

Since $\sec 0^\circ = \dfrac{1}{\cos 0^\circ} = \dfrac{1}{1} = 1$:

$$\sin 0^\circ \times \sec 0^\circ = 0 \times 1 = 0 = \tan 0^\circ$$

The identity $\tan\theta = \sin\theta \sec\theta$ holds, as it must for every angle where cosine is non-zero.

Example 5

Express $\tan 0^\circ$ in radians and evaluate $\tan 0$.

Since $0^\circ = 0$ radians, the angle is the same number in both units, and:

$$\tan 0 = 0$$

The radian form and the degree form name the same angle and the same value. A radian is just a different unit for measuring the same rotation.

Where Students Trip Up On Tan 0 Degrees

Mistake 1: Calling tan 0° undefined

Where it slips in: Right after learning that $\tan 90^\circ$ is undefined, when the two "edge" angles blur together.

Don't do this: Writing $\tan 0^\circ =$ undefined because "$0$ is a special angle." Students first meeting the tangent function often assume both ends of the first quadrant behave the same way.

The correct way: A tangent is undefined only when cosine is $0$. At $0^\circ$, cosine is $1$, not $0$, so $\tan 0^\circ = \dfrac{0}{1} = 0$. The undefined case is $90^\circ$, where cosine hits $0$.

Mistake 2: Confusing tan 0° with tan 45°

Where it slips in: Recall under time pressure, when the small first-quadrant values get swapped.

Don't do this: Writing $\tan 0^\circ = 1$. That is $\tan 45^\circ$, not $\tan 0^\circ$.

The correct way: Anchor on the pattern: tangent starts at $0$ ($0^\circ$), passes through $1$ ($45^\circ$), and races toward infinity near $90^\circ$. The value at the very start is $0$.

Mistake 3: Forgetting the calculator's angle mode

Where it slips in: Entering $\tan(0)$ is safe because $\tan 0$ is $0$ in both degree and radian mode, but the habit of not checking mode causes wrong answers on the next problem.

Don't do this: Trusting the screen for other angles without confirming whether it is set to degrees.

The correct way: Build the habit now. Confirm degree mode before entering any $\tan$ value, so the $0.15$-type surprise never happens on $\tan 30^\circ$ or $\tan 60^\circ$ later.

Key Takeaways

• Tan 0 degrees equals $0$, an exact whole number, because $\tan 0^\circ = \dfrac{\sin 0^\circ}{\cos 0^\circ} = \dfrac{0}{1}$.

• On the unit circle the $0^\circ$ point is $(1, 0)$, so tangent is $\dfrac{y}{x} = \dfrac{0}{1} = 0$.

• Tangent is undefined only where cosine is $0$ (at $90^\circ$), not where sine is $0$.

• In radians, $\tan 0^\circ = \tan 0 = 0$, and the value repeats every $180^\circ$.

To work through more special-angle values with a teacher, explore Bhanzu's trigonometry tutor, a high school math tutor, or flexible math classes online.

Practice These Before Moving On

1. Evaluate $7\tan 0^\circ - 2\tan 45^\circ$.

2. A road runs level for $50$ m at an angle of $0^\circ$. Use $\tan 0^\circ$ to find its vertical climb.

3. Show that $\tan 0^\circ + \cot 45^\circ = 1$.

Want a live Bhanzu trainer to walk through more tan 0 degrees problems? Book a free demo class.

Read More

• Tan 45 Degrees - value and proof

• Sin 0 Degrees - exact value

• Cos 0 Degrees - exact value

• Trigonometric functions - definitions and graphs