Trigonometry
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Trigonometry Reference Home Back to Thinkquest Contents
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Definitions of Trigonometric Functions Trigonometric Functions of Sums and Differences Inverse Trigonometric Functions
| 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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