Tangent Calculator
Enter an angle in degrees or radians to find its tangent, tan θ.
Updated
Tangent Calculator
Our Tangent Calculator finds tan θ for any angle in degrees or radians. Type the angle and the answer appears straight away to 10 decimal places, with the exact value for special angles (such as √3 for 60°), the angle in the other unit, the quadrant, and sin θ and cos θ. It tells you clearly when the tangent is undefined, at 90°, 270° and similar angles, instead of showing a huge meaningless number, and it gives tan 45° as exactly 1 rather than 0.9999999999999999.
Quick answer
The tangent of an angle is the opposite side divided by the adjacent side in a right triangle, which is the same as sin θ ÷ cos θ. For example, tan 45° = 1. Enter any angle above and the calculator shows tan θ instantly.
- Formula: tan θ = opposite ÷ adjacent = sin θ ÷ cos θ
- Key values: tan 0° = 0, tan 30° ≈ 0.5774, tan 45° = 1, tan 60° ≈ 1.7321
- Undefined: at 90°, 270° and every 90° + k × 180°, where cos θ = 0
- Range and period: any real number, repeating every 180° (π)
What Is Tangent?
Tangent is one of the three main trigonometric functions, with sine and cosine. In a right triangle it compares the two shorter sides: the one opposite the angle and the one next to it.
tan θ = sin θ ÷ cos θ
SOH: sin = opp / hyp
CAH: cos = adj / hyp
TOA: tan = opp / adj
Because tangent divides by cos θ, it has no value where cos θ = 0. Everywhere else it is defined, and unlike sine and cosine it is not limited to the range −1 to 1: it grows without limit as the angle approaches 90°.
Tangent Table of Common Angles
| Angle | Exact | Decimal |
|---|---|---|
| 0° (0) | 0 | 0 |
| 30° (π/6) | √3/3 | 0.5774 |
| 45° (π/4) | 1 | 1 |
| 60° (π/3) | √3 | 1.7321 |
| 90° (π/2) | undefined | undefined |
| 120° (2π/3) | −√3 | −1.7321 |
| 135° (3π/4) | −1 | −1 |
| 150° (5π/6) | −√3/3 | −0.5774 |
| 180° (π) | 0 | 0 |
√3/3 is the same number as 1/√3. From 180° onwards the values repeat, because tangent has a period of 180°.
The Tangent Graph
- Period: the curve repeats every 180° (π radians).
- Asymptotes: vertical lines at 90°, 270°, −90° and every 90° + k × 180°, where tan θ is undefined.
- Zeros: tan θ = 0 at 0°, 180° and every multiple of 180°.
- Always rising: between two asymptotes the curve climbs from −∞ to +∞.
- Odd function: tan(−θ) = −tan θ.
Tangent and Slope
The tangent of an angle is the slope of a line at that angle: how far it rises for every 1 unit across. That makes it the link between angles and gradients, pitches and grades:
| Angle | tan θ (slope) | As a grade |
|---|---|---|
| 2.86° | 0.05 | 5% |
| 4.76° | 0.0833 (1:12) | 8.3% |
| 26.57° | 0.5 (6/12 roof) | 50% |
| 45° | 1 | 100% |
A 100% grade is a 45° slope, not a vertical wall. To go from a slope back to an angle, use the inverse tangent: θ = tan−1(rise ÷ run).
Tangent Rules and Identities
- Quotient: tan θ = sin θ ÷ cos θ.
- Pythagorean: 1 + tan2θ = sec2θ.
- Negative angle: tan(−θ) = −tan θ.
- Half turn: tan(θ + 180°) = tan θ.
- Complement: tan(90° − θ) = cot θ = 1 ÷ tan θ.
- Double angle: tan 2θ = 2 tan θ ÷ (1 − tan2θ).
- Sum: tan(A + B) = (tan A + tan B) ÷ (1 − tan A tan B).
Worked Examples
Height of a tree. From 20 m away, the top of a tree is at a 35° angle of elevation:
= 20 × 0.7002
≈ 14.0 m
Angle of a ramp. A ramp rises 1 m over 12 m of ground:
≈ 4.76°
Radian inputs. tan(0.5) ≈ 0.5463, tan(1.2) ≈ 2.5722 and tan(2) ≈ −2.1850. 2 radians is about 115°, in quadrant II, so its tangent is negative.
Tangent in Excel and Code
| Tool | tan 45° |
|---|---|
| Excel / Google Sheets | =TAN(RADIANS(45)) |
| Python | math.tan(math.radians(45)) |
| JavaScript | Math.tan(45 * Math.PI / 180) |
| Scientific calculator | DEG mode, then tan 45 |
These functions expect radians, so =TAN(45) gives about 1.6198, the tangent of 45 radians. They also return 0.9999999999999999 for tan 45° and a huge number (about 1.63 × 1016) for tan 90° instead of an error, because π cannot be stored exactly.
Common Mistakes
- Wrong angle mode. tan 45 is 1 in degrees but about 1.6198 in radians.
- Trusting a huge result at 90°. A value like 1.63 × 1016 means the tangent is undefined there.
- Reading tan⁻¹ as 1/tan. tan−1(1) = 45° (arctan), while 1 ÷ tan θ is cot θ.
- Treating a grade as an angle. A 100% grade is 45°, and a 10% grade is about 5.71°, not 10°.
- Swapping the sides. Tangent is opposite ÷ adjacent; adjacent ÷ opposite is the cotangent.
Key Terms
- Opposite
- The side across from the angle θ.
- Adjacent
- The side next to θ that is not the hypotenuse.
- Asymptote
- A line the graph approaches but never reaches; tangent has them at 90° + k × 180°.
- Slope
- Rise ÷ run, equal to the tangent of the angle of the line.
- Cotangent
- The reciprocal of tangent: cot θ = 1 ÷ tan θ.
- Arctan
- The inverse of tangent, written tan⁻¹ or atan.
How to Use the Tangent Calculator
- Enter the angle. Type the angle θ, for example 45, or press one of the angle buttons (30°, 45°, 60° and 90° in degrees, π/6, π/4, π/3 and π/2 in radians). In radians you can type a decimal such as 0.5 or a multiple of π such as pi/4 or 3π/4.
- Choose degrees or radians. Pick Degrees (°) or Radians (rad). Typing 45° or 0.5 rad also sets the unit for that angle.
- Read and copy the result. tan θ appears as you type, to 10 decimal places, with the exact value where there is one (for example tan(60°) = √3 ≈ 1.7320508076). The table below shows the angle in the other unit, the quadrant, sin θ and cos θ. At 90°, 270° and similar angles the tangent is undefined and the calculator says so. Press Copy to copy the result, or Clear or Esc to start again.
Example: enter 135 in degrees and the result is −1, with the note that 135° is in quadrant II with a reference angle of 45°, and 135° = 3π/4 rad.
Who Can Use This Calculator?
Frequently Asked Questions (FAQs)
What is the tangent of an angle?
In a right triangle, the tangent of an angle θ is the side opposite θ divided by the side next to it: tan θ = opposite ÷ adjacent. For any angle it equals sin θ ÷ cos θ. It also gives the slope of a line: a line at angle θ to the horizontal rises tan θ units for every 1 unit across.
What is tan 0°, tan 30°, tan 45°, tan 60° and tan 90°?
tan 0° = 0, tan 30° = √3/3 ≈ 0.5774, tan 45° = 1 and tan 60° = √3 ≈ 1.7321. tan 90° is undefined, because cos 90° = 0 and you cannot divide by zero.
Why is tan 90° undefined?
tan θ = sin θ ÷ cos θ, and cos 90° = 0, so tan 90° would mean dividing 1 by 0. As θ gets closer to 90° the tangent grows without limit (tan 89.9° ≈ 573), and just past 90° it is hugely negative. The graph has a vertical asymptote there, and the same happens at 270°, −90° and every 90° + k × 180°.
Can tangent be greater than 1?
Yes. Unlike sine and cosine, tangent can be any real number. It is 1 at 45°, about 1.7321 at 60°, about 5.6713 at 80° and about 573 at 89.9°. Angles below 45° give values between 0 and 1.
When is tangent negative?
Tangent is positive in quadrants I and III (where sine and cosine have the same sign) and negative in quadrants II and IV. For example tan 120° ≈ −1.7321 and tan 135° = −1.
What is the period of the tangent function?
180°, or π radians, half the period of sine and cosine. Adding 180° gives the same value: tan 225° = tan 45° = 1.
How do you find an angle from a slope?
Use the inverse tangent (arctan): angle = tan−1(rise ÷ run). A 1 in 12 wheelchair ramp has a slope of 1/12, so its angle is tan−1(1/12) ≈ 4.76°. A road with a 5% grade rises at tan−1(0.05) ≈ 2.86°.
Is tan⁻¹ the same as 1/tan?
No. tan−1(x) usually means arctan, the angle whose tangent is x: tan−1(1) = 45°. The reciprocal 1 ÷ tan θ is a different function called cotangent, cot θ: cot 45° = 1 and cot 60° ≈ 0.5774.
Why does my calculator or code give 0.9999999999999999 for tan 45°?
Programs work in radians and cannot store π exactly, so Math.tan(Math.PI / 4) in JavaScript and math.tan(math.pi / 4) in Python return 0.9999999999999999, and tan(π/2) gives about 1.63 × 1016 instead of an error. This calculator works in degrees for special angles, so tan 45° is exactly 1 and tan 90° is reported as undefined.
Is the Tangent Calculator free?
Yes. It is completely free, needs no sign-up, and can be used as many times as you like.
Try our Tangent Calculator to find tan θ for any angle in degrees or radians instantly.