Tangent Calculator — tan(x) in Degrees or Radians
tan(x)
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Angle in Degrees
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Angle in Radians
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Understanding the Tangent Function
The tangent function equals sin(x)/cos(x), or equivalently, the ratio of the opposite side to the adjacent side in a right triangle. Unlike sine and cosine, tangent has no maximum or minimum value and has vertical asymptotes where cosine equals zero (at 90, 270 degrees, etc.).
Key values: tan(0) = 0, tan(30) = 1/sqrt(3), tan(45) = 1, tan(60) = sqrt(3). The tangent function has a period of 180 degrees (pi radians), half that of sine and cosine.
Tangent is used to calculate slopes, grades, and angles of elevation or depression. In surveying, the tangent of the angle of elevation times the distance gives the height of an object.
Frequently Asked Questions
When is tangent undefined?
Tangent is undefined at 90, 270, 450 degrees, etc. (odd multiples of 90). At these angles, cosine equals zero, making the ratio sin/cos undefined. These are vertical asymptotes.
What is tan(45)?
tan(45 degrees) = 1 exactly. This is because at 45 degrees, the opposite and adjacent sides of a right triangle are equal, making their ratio 1.
Is tangent periodic?
Yes, with period 180 degrees (pi radians). This means tan(x + 180) = tan(x) for all x where tangent is defined.