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
| Half Angle Formulas Half angle formulas allow us to compute the sine and cosine functions of 1/2 of the angle, when the cosine of the angle is known. Start with Cos(2X) = Cos2X - Sin2X Substitute Sin2X = 1 - Cos2X Cos2X = Cos2X - 1 + Cos2X = 2Cos2X - 1 Add 1 to both sides 1 + Cos2X = 2Cos2X Divide by 2 (1 + Cos2X)/2 = Cos2X Define 2X = Y X = Y/2 (1 + CosY)/2 = Cos2(Y/2) This equation gives Cos(Y/2) when CosY is known. For Y = 30° and the value Cos30° = √3/2. (1 + √3/2)/2 = Cos215° Cos15° = .9659
To find Sin(Y/2), use: Sin2(Y/2) = 1 - Cos2(Y/2) = 1 - (1 + CosY)/2 Sin2(Y/2) = (1 - CosY)/2 For Y = 30°, Sin2(15°) = (1 - √3/2)/2 Sin15° = .2588
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