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)
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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))

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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
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See Also

Functions Overview, Functionstree, Settings window, System functions