一條簡單Compound Angles問題

Tieria

2008-12-14 3:03 am

Solve the following equation for (0ﾟ小於或等於θ小於或等於360ﾟ). Give your answers correct to the nearest 0.1ﾟ

Sin^4θ+ cos^4θ= sin2θ

答案是23.5ﾟ, 66.5ﾟ, 203.5ﾟ, 246.5ﾟ。

回答 (1)

Uncle Michael

2008-12-14 4:01 am

✔ 最佳答案

sin4θ + cos4θ = sin2θ

(sin2θ)2 + (cos2θ)2 = sin2θ

(sin2θ)2 + 2sin2θcos2θ + (cos2θ)2 = sin2θ + 2sin2θcos2θ

(sin2θ + cos2θ)2 = sin2θ + 2(sin2θcos2θ)2

(1)2 = sin2θ + 2[(sin2θ)/2]2

(1/2)sin22θ + sin2θ - 1 = 0

sin22θ + 2sin2θ - 2 = 0

sin2θ = {-2+√[22-4(1)(-2)]}/2 or {-2-√[22-4(1)(-2)]}/2

sin2θ = 0.7321 or sin2θ = -2.7321(rejected)

2θ = 47o or 2θ = 180o-47o or 2θ = 360o+47o or 2θ = 360o+180o-47o

2θ = 47o or 2θ = 133o or 2θ = 407o or 2θ = 493o

Hence, θ = 23.5o or θ = 66.5o or θ = 203.5o or θ = 246.5o
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