Math
Tangent Calculator
Tangent is the ratio opposite / adjacent in a right triangle and equals sin θ / cos θ, which is why it is undefined wherever cosine is zero.
This tangent calculator evaluates tan θ in degrees or radians with exact answers such as tan 60° = √3 and tan 30° = √3/3, solves tan θ = x for every angle, and finds a missing right-triangle side or angle.
Type a number, a fraction like 180/7, a π expression like 5π/6, or degrees-minutes-seconds like 30° 15′ 30″. Negative and large angles are fine.
Try an example
Result
tangent result
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- Angle in degrees
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- Angle in radians
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- Quadrant
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- Reference angle
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- Reciprocal (cot)
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- Cofunction / ratio
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More details (1 more)
- More
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Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
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Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What tangent measures
Tangent is the How steep a line is: how much y changes for each step right in x. of the terminal side: rise over run, or sin θ divided by cos θ. It grows without bound as the angle approaches 90°, because the run (cos θ) shrinks to zero.
Exact tangent values
| θ | tan θ | Decimal |
|---|---|---|
| 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 | — |
| 120° (2π/3) | -√3 | -1.7321 |
| 135° (3π/4) | -1 | -1 |
| 180° (π) | 0 | 0 |
| 225° (5π/4) | 1 | 1 |
Four ways to use this calculator
| Mode | You enter | You get |
|---|---|---|
| Find tan θ | An angle in any unit | tan θ exact and decimal, cot θ, cot of the complement, quadrant, reference angle |
| Find θ (arctan) | Any real number | Principal angle (-90° to 90°), both angles in [0°, 360°), general solution |
| Right-triangle side | An acute angle and the opposite or adjacent side | The other of the two, plus the hypotenuse and other angle |
| Right-triangle angle | Opposite and adjacent sides | The angle, hypotenuse, and other angle |
How to read the answer
- tan θ is positive in One of the four regions the x- and y-axes cut the coordinate plane into. I and III and negative in Quadrants II and IV.
- Tangent repeats every 180°, so tan θ = x always has exactly two solutions in one full turn.
- undefined means cos θ = 0 at that angle (90°, 270°, and so on); the graph has a vertical asymptote there.
- The cofunction line shows cot(90° − θ), which always equals tan θ.
Common mistakes and edge cases
- tan 45 in An angle unit where a full turn is 2π instead of 360 degrees. mode is 1.62, not 1. Check the unit selector.
- tan 90° is not infinity or zero; it is undefined. The calculator says so instead of printing a huge number.
- arctan only returns angles between -90° and 90°. For a Quadrant II or III angle add 180°.
- Tangent has no upper limit, so an inverse-mode input like 1000 is valid and gives an angle just under 90°.
Worked examples
tan 60°
tangent result
√3 (1.732051)
Quadrant II tangent
tan 135° = -1
tangent result
-1
Undefined tangent
tan 90° divides by zero
tangent result
undefined
tan in radians
tan(π/6) = √3/3
tangent result
√3/3 (0.57735)
Solve tan θ = 1
tangent result
45° = π/4
Solve tan θ = -√3
tangent result
-60° = -π/3
Find the opposite side
θ = 30°, adjacent = 10
tangent result
Opposite side = 5.773503
Find the adjacent side
θ = 45°, opposite = 6
tangent result
Adjacent side = 6
Angle from two legs
opposite 3, adjacent 4
tangent result
θ = 36.8699°
Angle of 90° in triangle mode
tangent result
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Frequently asked questions
Why is tan 90 degrees undefined?+
tan θ = sin θ / cos θ and cos 90° = 0. Dividing by zero has no value, so tan 90° is undefined and the graph of tangent has a vertical asymptote at 90°.
What is tan 45 degrees?+
tan 45° = 1, because a 45-45-90 triangle has equal legs, so opposite / adjacent = 1.
Why does tan θ = 1 have two solutions between 0° and 360°?+
Tangent repeats every 180°, so 45° and 45° + 180° = 225° both have tangent 1. Sine and cosine repeat every 360°, which is why their equations pair up differently.
How is tangent related to slope?+
A line that makes angle θ with the positive x-axis has slope tan θ. A 45° line has slope 1; a 60° line has slope √3 ≈ 1.732.
How do I find a side with tangent?+
tan θ = opposite / adjacent. Know the adjacent side? opposite = adjacent × tan θ. Know the opposite side? adjacent = opposite / tan θ.
What is the inverse of tangent?+
Arctangent (tan⁻¹). It returns the angle between -90° and 90° whose tangent is the given value; add 180° to get the other solution in one turn.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026