F.4 TRIGONOMETRY M2

2011-03-09 1:30 am

1. PROVE THATsin(32+a)cos(58-b)+cos(32+a)sin(58-b) = cos(a-b)

2.PROVE THAT sin(2a+b)/sina - 2cos(a+b) = sinb/sina

3. PROVE THAT tan(45+θ)+tan(45-θ) = 2/cos^2θ-sin^2θ

回答 (1)

myisland8132

2011-03-09 1:54 am

✔ 最佳答案

1 Let x = 32 + a, y = 58 - b

sin(32+a)cos(58-b)+cos(32+a)sin(58-b)

= sinxcosy+cosxsiny

= sin(x + y)

= sin(32 + a + 58 - b)

= sin(90 + a - b)

=cos(a - b)

2 sin(2a + b)/sina - 2cos(a + b)

= (sin2acosb + cos2asinb)/sina - 2cos(a + b)

= (2sinacosacosb + sinb - 2sin^2asinb)/sina - 2cos(a + b)

= (2sinacos(a + b) + sinb)/sina - 2cos(a + b)

= sib/sina

3 tan(45+θ)+tan(45-θ)

= (tan45 + tanθ)/(1 - tan45tanθ) + (tan45 - tanθ)/(1 + tan45tanθ)

= (1 + tanθ)/(1 - tanθ) + (1 - tanθ)/(1 + tanθ)

= (cosθ + sinθ)/(cosθ - sinθ) + (cosθ - sinθ)/(cosθ + sinθ)

= 2/(cos^2θ - sin^2θ)

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