Trigonometric Functions
For a specific angle, the trigonometric functions are defined as ratios of the lengths of the sides of a right triangle.
Angles can be measured in radians, degrees or gradians. One complete circle contains 2*Pi radians, 360 degrees or 400 gradians.
Before doing calculations with trigonometric functions it's important to consider if you're using radians, degrees or gradians, to be sure of obtaining the correct results. With SpeQ you can adjust the Angles mode in the statusbar, or in the Settings window.
Overview
You can use the following built-in trigonometric functions.| Function | Description |
|---|---|
| Sin(x) | Sine of x |
| Cos(x) | Cosine of x |
| Tan(x) | Tangent of x |
| ASin(x) | Inverse sine of x, or Sin-1(x) |
| ACos(x) | Inverse cosine of x, or Cos-1(x) |
| ATan(x) | Inverse tangent of x, or Tan-1(x) |
| ATan2(y, x) | Four-quadrant inverse tangent of the real parts of x and y. Computes the angle of the vector (x,y) in radians, in the range [-pi, Pi]. |
| Csc(x) | Cosecant of x, Defined as: Csc(x) = 1/Sin(x) |
| Sec(x) | Secant of x, Defined as: Sec(x) = 1/Cos(x) |
| Cot(x) | Cotangent of x, Defined as: Cot(x) = 1/Tan(x) |
Trigonometric Identities
The trigonometric functions satisfy a number of well-known mathematical identities. The most important identities are listed below.| Hyperbolic Identities |
|---|
|
General identities Tan(x) = Sin(x) / Cos(x) Cot(x) = Cos(x) / Sin(x) Sin(-x) = -Sin(x) Cos(-x) = Cos(x) Sin(x + 2*Pi) = Sin(x) Cos(x + 2*Pi) = Cos(x) Pythagorean identities Sin2(x) + Cos2(x) = 1 Tan2(x) + 1 = Sec2(x) 1 + Cot2(x) = Csc2(x) Addition identities Sin(x + y) = Sin(x)*Cos(y) + Cos(x)*Sin(y) Cos(x + y) = Cos(x)*Cos(y) - Sin(x)*Sin(y) Subtraction identities Sin(x - y) = Sin(x)*Cos(y) - Cos(x)*Sin(y) Cos(x - y) = Cos(x)*Cos(y) + Sin(x)*Sin(y) Double-angle identities Sin(2*x) = 2*Sin(x)*Cos(x) Cos(2*x) = Cos2(x) - Sin2(x) Cos(2*x) = 2*Cos2(x) - 1 Cos(2*x) = 1 - 2*Sin2(x) Half-angle identities Cos2(x) = (1 + Cos(2*x)) / 2 Sin2(x) = (1 - Cos(2*x)) / 2 Product identities Sin(x)*Cos(y) = 1/2 * (Sin(x+y) + Sin(x-y)) Cos(x)*Cos(y) = 1/2 * (Cos(x+y) + Cos(x-y)) Sin(x)*Sin(y) = 1/2 * (Cos(x-y) - Cos(x+y)) |
Examples
'Examples of using Trigonometric Functions
'Set angle mode to radians. A circle contains 2*Pi radians.
Angles = Rad
Angles set to Radians
Sin(0.25 * Pi)
Ans = 0.7071067812
Cos(0.25 * Pi)
Ans = 0.7071067812
Tan(1)
Ans = 1.5574077247
ASin(0.5)
Ans = 0.5235987756
Sin(Ans)
Ans = 0.5
Sec(0.25 * Pi) ^ 2
Ans = 2
'Set angle mode to degrees. A circle contains 360 degrees.
Angles = Deg
Angles set to Degrees
Sin(45)
Ans = 0.7071067812
Sqrt(2) / 2
Ans = 0.7071067812
'Define a variable and use it in a calculation.
MyAngle = 22.5
MyAngle = 22.5
Sin(MyAngle) ^ 2 + Cos(MyAngle) ^ 2
Ans = 1
Sin(22.5) ^ 2 + Cos(22.5) ^ 2
Ans = 1