Hyperbolic Functions Calculator - Six Functions, One Input
Use this free hyperbolic functions calculator to evaluate sinh, cosh, tanh, coth, sech, and csch from a real input in degrees or radians.
Hyperbolic Functions Calculator
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What Is a Hyperbolic Functions Calculator?
A hyperbolic functions calculator is a math tool that takes one real input x and returns the six standard hyperbolic functions - sinh, cosh, tanh, coth, sech, and csch - in a single result panel. You type a real number, choose radians or degrees, and read the six values that the exponential definition produces. A worked example sits next to the formulas so you can verify the result by hand.
- • Checking a homework or textbook identity: Students can plug in an x value and read all six outputs side by side, faster than calling Math.sinh, Math.cosh, and Math.tanh separately.
- • Verifying a value before a downstream step: Engineers, physicists, and statisticians who use the Gudermannian, the logistic function, or hyperbolic PDE solutions can confirm the hyperbolic value of x before substituting it into a follow-up formula.
- • Plotting the hyperbolic family of curves: Readers can tabulate values across an x range and see cosh^2 - sinh^2 = 1 hold for every row, which makes the geometric interpretation of (cosh x, sinh x) as a point on the right branch of the unit hyperbola easy to draw.
Because the hyperbolic functions are built from e^x and e^(-x), readers who need to express a number in scientific notation first can keep the same number in the Exponential Notation Calculator and read the mantissa and exponent side by side.
How the Hyperbolic Functions Calculator Works
The tool reads the input x, converts it to radians if the unit is degrees, and evaluates the six hyperbolic functions from the exponential definition. The formulas sit next to the result panel so the reader can follow the rule that produced each output.
- x (input): Real input at which to evaluate the six functions. Accepted in radians or degrees; normalised to radians internally.
- e^x and e^(-x) (exponentials): The two exponential values the formula needs. Computed with Math.exp, so the result matches the textbook value of (e^x - e^(-x))/2 and (e^x + e^(-x))/2.
- Reciprocals (coth, sech, csch): Computed from sinh and cosh by direct division. When x = 0 the result panel shows Undefined for the two rows with a true singularity.
The exponential form has the same shape for sinh and cosh, so once the calculator knows e^x and e^(-x) it can write the half-difference and the half-sum in one pass and then read tanh, coth, sech, and csch off the same two numbers. This makes the identity cosh^2(x) - sinh^2(x) = 1 a one-line check on the result.
Worked example: x = 1, six hyperbolic values in one panel
x = 1, unit Radians.
e^1 ≈ 2.7182818285, e^(-1) ≈ 0.3678794412. sinh(1) ≈ 1.1752011936. cosh(1) ≈ 1.5430806348. tanh(1) ≈ 0.7615941560.
sinh(1) ≈ 1.175201, cosh(1) ≈ 1.543081, tanh(1) ≈ 0.761594, coth(1) ≈ 1.313035, sech(1) ≈ 0.648054, csch(1) ≈ 0.850918.
Identity check: cosh^2(1) - sinh^2(1) ≈ 1.00000.
According to Weisstein, Eric W. "Hyperbolic Functions," MathWorld, the six hyperbolic functions are defined by sinh x = (e^x - e^(-x))/2, cosh x = (e^x + e^(-x))/2, tanh x = sinh x / cosh x, and coth, sech, csch as the three reciprocals.
Readers who want to take one of the outputs and apply it to the most common cosh application, the hanging-chain curve, can pass the hyperbolic cosine into the Catenary Curve Calculator, which evaluates y = a cosh(x/a) and the arc length of a uniform chain or cable.
Key Concepts Behind the Hyperbolic Functions
Four short definitions keep the formulas honest. The first three are the exponential building blocks, and the fourth is the algebraic identity that lets the reader check the result panel against a textbook table.
sinh from the half-difference of exponentials
sinh(x) = (e^x - e^(-x))/2 is the half-difference of the two exponentials. It is odd, so sinh(-x) = -sinh(x), and it passes through the origin: sinh(0) = 0.
cosh from the half-sum of exponentials
cosh(x) = (e^x + e^(-x))/2 is the half-sum of the two exponentials. It is even and has a global minimum of 1 at x = 0, which is the basis for cosh^2(x) - sinh^2(x) = 1.
tanh as the ratio, plus coth, sech, csch as reciprocals
tanh(x) = sinh(x) / cosh(x) always sits strictly between -1 and 1. coth, sech, and csch are the reciprocals; coth and csch have true singularities at x = 0.
The identity cosh^2(x) - sinh^2(x) = 1
This identity is the hyperbolic analogue of the Pythagorean identity sin^2(x) + cos^2(x) = 1. The worked example confirms it returns 1.000000 for x = 1.
Readers who want to extend the hyperbolic family into the complex plane and verify the identity sinh(ix) = i sin(x) can pass the input into the Complex Number Calculator, which evaluates the same expressions in the complex plane and returns the real and imaginary parts separately.
How to Use the Hyperbolic Functions Calculator
Type a real number, pick its unit, and read the six hyperbolic values. The result panel updates as you type, so there is no submit step to remember.
- 1 Enter the real input x: Type the value of x in the Input x box. The default is 1.0, which gives non-trivial results for all six functions.
- 2 Pick radians or degrees: Choose Radians for a pure-math input or Degrees if your source uses degree units such as 180.
- 3 Read the three core functions: The first three result rows give sinh, cosh, and tanh, which are the building blocks for the rest of the panel.
- 4 Read the three reciprocal functions: The next three rows give coth, sech, and csch, computed from sinh and cosh by direct division.
- 5 Watch the x = 0 row: When x = 0, coth and csch are genuinely undefined. The result panel reports Undefined for those two rows and shows the four finite values.
- 6 Cross-check with the identity: The result panel values confirm cosh^2(x) - sinh^2(x) = 1 to six-decimal display precision.
A reader needs all six hyperbolic values of x = 2 for a logistic regression derivation. They type 2, leave the unit on Radians, and read sinh(2) ≈ 3.626860, cosh(2) ≈ 3.762196, tanh(2) ≈ 0.964028, coth(2) ≈ 1.037315, sech(2) ≈ 0.265802, csch(2) ≈ 0.275721. The identity check confirms cosh^2(2) - sinh^2(2) ≈ 1.00000.
Readers who need to convert a hyperbolic input that lives in one angle system into the other can pass it through the Radians to Degrees Calculator, which gives the exact radian or degree equivalent for any real value.
Benefits of Using the Hyperbolic Functions Calculator
A short list of what the tool does well, and what it is not designed to do, helps you put the result in the right place.
- • All six functions in one panel: The calculator returns sinh, cosh, tanh, coth, sech, and csch at the same time, so a single read covers the six hyperbolic identities and the three reciprocals.
- • Radians and degrees in the same input box: Choose either unit; the calculator normalises to radians internally so the result panel always shows the hyperbolic value of the radian-equivalent input.
- • Identity check built in: The worked example confirms cosh^2(x) - sinh^2(x) = 1, so the result panel doubles as a sanity check for any downstream formula.
- • Singularities handled honestly: When x = 0, coth and csch are genuinely undefined (division by zero), and the tool reports Undefined for those two rows.
Readers working on the differential equations side, where hyperbolic functions and Bessel functions tend to appear together in the same PDE solutions, can continue the analysis with the Bessel Function Calculator, which evaluates J_n(x) and its recurrence partners for integer order and real argument.
Factors That Affect Your Hyperbolic Functions Result
The tool is honest about which factors change the number, which only change the unit, and which it cannot see at all.
Choice of radians versus degrees
Radians and degrees are scaled versions of the same number. The calculator normalises to radians, so the hyperbolic value of 180 degrees equals the hyperbolic value of pi radians.
Sign of x and parity of each function
sinh, tanh, coth, and csch are odd, so they flip sign for negative x. cosh and sech are even and stay positive. The calculator uses the full exponential form, so the parity rules are correct for any real x.
Singularities at x = 0 for coth and csch
coth(0) and csch(0) are genuinely undefined. The tool reports Undefined for those two rows when x = 0 and still shows the four finite values in the same panel.
Magnitude of the six values across the input range
The input box accepts -10 to 10. In that range, |sinh(10)| and cosh(10) reach roughly 11013, tanh stays in (-1, 1), and sech and csch shrink toward 0. e^10 ≈ 22026 still fits the 64-bit float range with room to spare, so no JavaScript overflow occurs.
Floating-point precision
Six-decimal display precision matches standard textbook tables. Higher precision is available by computing offline.
- • The tool returns the principal hyperbolic value of the radian-equivalent input. It does not solve for x given a known hyperbolic value; use the inverse hyperbolic formulas on paper or in a separate script for that direction.
- • The Undefined row for coth and csch at x = 0 is a signal that the formula has no real answer at that input, not a JavaScript artifact.
According to Wikipedia, "Hyperbolic functions", the identity cosh^2(x) - sinh^2(x) = 1 mirrors the Pythagorean identity for the trigonometric functions, and the six inverse hyperbolic functions are arcsinh x = ln(x + sqrt(x^2 + 1)), arcosh x = ln(x + sqrt(x^2 - 1)), and artanh x = (1/2) ln((1+x)/(1-x)).
Readers who need to convert the input between the two angle systems or to minutes, seconds, or revolutions can move it through the Angle Converter Calculator, which handles every common angle unit on the same page.
Frequently Asked Questions
Q: What are the six hyperbolic functions?
A: The six hyperbolic functions are sinh, cosh, tanh, coth, sech, and csch. The first three are built directly from exponentials: sinh x = (e^x - e^(-x))/2, cosh x = (e^x + e^(-x))/2, and tanh x = sinh x / cosh x. The last three are the reciprocals: coth = cosh / sinh, sech = 1 / cosh, csch = 1 / sinh.
Q: How do you calculate sinh, cosh, and tanh from exponentials?
A: Use the exponential definition. Compute e^x and e^(-x), then take sinh x = (e^x - e^(-x))/2, cosh x = (e^x + e^(-x))/2, and tanh x = sinh x / cosh x. The result panel does this for any real x in radians or degrees, so you do not have to call exp twice by hand.
Q: What is the identity cosh^2(x) - sinh^2(x) = 1?
A: cosh^2(x) - sinh^2(x) = 1 is the hyperbolic analogue of the Pythagorean identity sin^2(x) + cos^2(x) = 1. It holds for every real x, and the result panel's worked example confirms it returns 1.000000 for x = 1, so the identity is the quickest way to check that the calculator is returning the right values.
Q: What is the value of sinh(0) and cosh(0)?
A: sinh(0) = 0 and cosh(0) = 1, just like the trigonometric sine and cosine. tanh(0) = 0 and sech(0) = 1 follow directly. coth(0) and csch(0) are genuinely undefined because they require division by sinh(0) = 0, so the result panel reports Undefined for those two rows at x = 0.
Q: What are the inverse hyperbolic functions?
A: The inverse hyperbolic functions are arcsinh, arcosh, artanh, arcoth, arsech, and arcsch. arcsinh x = ln(x + sqrt(x^2 + 1)) is defined for all real x, arcosh x = ln(x + sqrt(x^2 - 1)) is defined for x >= 1, and artanh x = (1/2) ln((1+x)/(1-x)) is defined for |x| < 1. This calculator returns the forward six; use the inverse formulas in a separate script for the reverse direction.
Q: Why are hyperbolic functions not periodic?
A: Hyperbolic functions are not periodic because the exponentials e^x and e^(-x) grow and decay monotonically as x moves away from zero. sin and cos are periodic because they are built from the imaginary exponentials e^(ix) and e^(-ix), which cycle around the unit circle. The hyperbolic family instead parametrises the right branch of the unit hyperbola, which never revisits the same point.