A.Maths Trigonometric Functions of Compound Angles

2007-12-26 8:51 pm

Suppose sin (X+B) = k sin (X-B) and k<>1

then tan X = [ tan B (k+1) ] / (k-1)

Hence solve the equation

sin( θ + π/3 ) = 3 sin [ (2π/3) + θ ] , where 0≦θ<2π

(Give the answers correct to 4 decimal places)

更新1:

The answers should be 0.7137 and 3.8553

回答 (1)

myisland8132

2007-12-26 11:02 pm

✔ 最佳答案

Suppose sin (X+B) = k sin (X-B) and k<>1

then tan X = [ tan B (k+1) ] / (k-1)

Hence solve the equation

sin( θ + π/3 ) = 3 sin [ (2π/3) + θ ] , where 0≦θ<2π
sin (X+B) = k sin (X-B)
sinXcosB+cosXsinB=k(sinXcosB-cosXsinB)
(1-k)sinXcosB=-(k+1)cosXsinB
(k-1)sinXcosB=(k+1)cosXsinB
tan X = [ tan B (k+1) ] / (k-1)
-sin( θ + π/3 ) = 3 sin [ (2π/3) + θ ] , where 0≦θ<2π
-sin( π+θ + π/3 ) = 3 sin [ π-(π/3) + θ ] , where 0≦θ<2π
sin( π+θ + π/3 ) = -3 sin [ π-(π/3) + θ ] , where 0≦θ<2π
Let X=π+θ, B=π/3,k=-3
The equation becomes
tan π+X = [ tan (π/3) (-3+1) ] / (-3-1)
tan X = √3 / 2
X=40.8934 or 220.8934 (correct to 4 decimal places)

2007-12-26 15:46:16 補充：
40.8934 or 220.8934 度即是0.7137 和 3.8553 弧度上面個位計錯數了

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