Trigonometric Functions of Sums and Differences
Often, we need to study more complex formulas with the trigonometric functions. Sin(X) is easy to solve, but now we will learn Sin(X + Y) where the variable Y is added to X. The same is true for Cos(X + Y).
Sin(X + Y) = Sin(X)Cos(Y) + Cos(X)Sin(Y)
Cos(X + Y) = Cos(X)Cos(Y) - Sin(X)Sin(Y)
Cos(X - Y) = Cos(X)Cos(Y) + Sin(X)Sin(Y)
Example: Use the trig functions for X = 30° and Y = 45° to find the Sin75°.
Sin 30° = 1/2
Sin 45° = √2/2
Cos 30° = √3/2
Cos 45° =√ 2/2
Sin 75° = Sin(30° + 45°) =
Sin(30°)Cos(45°) + Cos(30°)Sin(45°) =
(1/2)(√2/2) + (√3/2)(√2/2) = (√2 + √6)/4
The sum and difference formulas for the tangent function is a little different.
Tan(X + Y) = (TanX + TanY)/(1 - TanXTanY)
Tan(X - Y) =( TanX - TanY)/(1 + TanXTanY)
To further understand this formula, let us look at its proof:
Tan(X + Y) = Sin(X + Y)/Cos(X + Y) = (SinXCosY + CosXSinY)/(CosXCosY - SinXSinY)
divide numerator and denominator by CosXCosY:
(SinXCosY)/(CosXCosY) + CosXSinY/(CosXCosY) =
CosXCosY/(CosXCosY) - SinXSinY/(CosXCosY)
(TanX + TanY)/(1 - TanXTanY)
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