BIZOMAT MAKEOS · Mathematics
Trigonometric and Involute Functions
What are the trigonometric, versed and involute functions of this angle, or the angle for an involute value?
Sine, cosine, tangent, cotangent, secant, cosecant, versed sine and cosine, and the involute function of an angle, or the angle whose involute is a given value (inverse involute) for gear work.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Calculate | choice | Find functions of an angle, or the angle for an involute value. |
| Angle | ° | Angle in degrees. |
| Involute value | Involute function value (inv α = tan α − α). |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Calculate
- What it is
- Whether to work out the trigonometric and involute functions of a known angle, or to find the angle that gives a known involute value (used in gear and spline work).
- Units and format
- Choose one of 2 options from the list: Functions of an angle; Angle from an involute value.
- Example
- The form opens with “Functions of an angle” selected as a worked example. Choose the option that matches your case.
- Get it right
- Only the Angle input is used in the first mode and only the Involute value in the second.
What this means: Angle
- What it is
- The angle whose sine, cosine, tangent, versed functions and involute you want, such as a gear pressure angle or a taper angle.
- Units and format
- Enter a number in °: degrees of angle. The form accepts -3600° to 3600°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Type degrees, not radians. Used only when calculating functions of an angle.
What this means: Involute value
- What it is
- A known involute function value, inv α = tan α − α with α in radians, for which you want the angle; it often comes from a gear calculation such as a measurement over pins.
- Units and format
- Enter a plain number with no unit. The form accepts 0 to 10. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.014904 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Used only in the angle-from-involute mode; the angle found must be below 90°.
Method and formulas
- inv α = tan α − α (α in radians).
- Versed sine = 1 − cos α; versed cosine = 1 − sin α.
- Inverse involute solved by Newton’s method.
- Example: inv 20° = 0.01490438.
Limitations
- The involute is defined for 0° ≤ α < 90°.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Replace printed trig and involute tables
- Find pressure angles from involute values in gear calculations
- Check versed sine for chord heights
Who it is for
- Engineers
- Technicians
- Engineering students
What you get
- Sine, cosine, tangent, cotangent, secant, cosecant
- Versed sine and versed cosine
- Involute, or the angle from an involute
Interpreting the result
- Versed sine × radius gives the height of an arc above its chord.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 95–114. Topic reference; no source tables or recommended settings are reproduced.
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