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ss_6_3

# ss_6_3 - Section 6.3 Double-Angle and Half-Angle Formulas...

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Section 6.3 Double-Angle and Half-Angle Formulas This section contains important trigonometric identities for finding the trigonometric functions of double angles and half angles. Double-Angle Formulas sin 2 θ = 2 sin θ cos θ (sin( θ + θ ) = sin θ cos θ + cos θ sin θ ) cos 2 θ = cos 2 θ - sin 2 θ (cos( θ + θ ) = cos θ cos θ - sin θ sin θ ) cos 2 θ = 1 - 2 sin 2 θ cos 2 θ = 2 cos 2 θ - 1 Example. If sin θ = 4 5 and π 2 < θ < π , find sin 2 θ . Solution. cos θ = - 3 5 . Hence sin 2 θ = 2 sin θ cos θ = 2( 4 5 )( - 3 5 ) = - 24 25 tan 2 θ = 2 tan θ 1 - tan 2 θ (tan( θ + θ ) = tan θ +tan θ 1 - tan θtanθ ) Since cos 2 θ = 1 - 2 sin 2 θ = 2 cos 2 θ - 1, we have the following: sin 2 θ = 1 - cos 2 θ 2 cos 2 θ = 1+cos 2 θ 2 tan 2 θ = 1 - cos 2 θ 1+cos 2 θ Example. cos 4 θ = (cos 2 θ ) 2 = ( 1+cos 2 θ 2 ) 2 = 1 4 (1 + 2 cos2 θ + cos 2 2 θ ) = 1 4 + 1 2 cos 2 θ + 1 4 ( 1+cos 4 θ 2 ) = 3 8 + 1 2 cos 2 θ + 1 8 cos 4 θ Half-Angle Formulas sin α 2 = ± q 1 - cos α 2 cos α 2 = ± q 1+cos α 2 tan α 2 = ± q 1 - cos α 1+cos α

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ss_6_3 - Section 6.3 Double-Angle and Half-Angle Formulas...

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