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人気のある三角関数 >

sin^2(a-pi/8)= 2/3

解

sin^{2} (a - \frac{π}{8}) = \frac{2}{3}

解

a = 0.95531 \dots + 2 πn + \frac{π}{8}, a = π - 0.95531 \dots + 2 πn + \frac{π}{8}, a = - 0.95531 \dots + 2 πn + \frac{π}{8}, a = π + 0.95531 \dots + 2 πn + \frac{π}{8}

+1

度

a = 77.23561 \dots^{\circ} + 36 0^{\circ} n, a = 147.76438 \dots^{\circ} + 36 0^{\circ} n, a = - 32.23561 \dots^{\circ} + 36 0^{\circ} n, a = 257.23561 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

sin^{2} (a - \frac{π}{8}) = \frac{2}{3}

置換で解く

sin^{2} (a - \frac{π}{8}) = \frac{2}{3}

仮定:

sin (a - \frac{π}{8}) = u

u^{2} = \frac{2}{3}

u^{2} = \frac{2}{3} : u = \frac{2}{3}, u = - \frac{2}{3}

u^{2} = \frac{2}{3}

x^{2} = f (a)

の場合, 解は

x = f (a), - f (a)

u = \frac{2}{3}, u = - \frac{2}{3}

代用を戻す

u = sin (a - \frac{π}{8})

sin (a - \frac{π}{8}) = \frac{2}{3}, sin (a - \frac{π}{8}) = - \frac{2}{3}

sin (a - \frac{π}{8}) = \frac{2}{3}, sin (a - \frac{π}{8}) = - \frac{2}{3}

sin (a - \frac{π}{8}) = \frac{2}{3} : a = arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}, a = π - arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

sin (a - \frac{π}{8}) = \frac{2}{3}

三角関数の逆数プロパティを適用する

sin (a - \frac{π}{8}) = \frac{2}{3}

以下の一般解

sin (a - \frac{π}{8}) = \frac{2}{3}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

a - \frac{π}{8} = arcsin (\frac{2}{3}) + 2 πn, a - \frac{π}{8} = π - arcsin (\frac{2}{3}) + 2 πn

a - \frac{π}{8} = arcsin (\frac{2}{3}) + 2 πn, a - \frac{π}{8} = π - arcsin (\frac{2}{3}) + 2 πn

解く

a - \frac{π}{8} = arcsin (\frac{2}{3}) + 2 πn : a = arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

a - \frac{π}{8} = arcsin (\frac{2}{3}) + 2 πn

\frac{π}{8}

を右側に移動します

a - \frac{π}{8} = arcsin (\frac{2}{3}) + 2 πn

両辺に

\frac{π}{8}

を足す

a - \frac{π}{8} + \frac{π}{8} = arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

簡素化

a = arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

a = arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

解く

a - \frac{π}{8} = π - arcsin (\frac{2}{3}) + 2 πn : a = π - arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

a - \frac{π}{8} = π - arcsin (\frac{2}{3}) + 2 πn

\frac{π}{8}

を右側に移動します

a - \frac{π}{8} = π - arcsin (\frac{2}{3}) + 2 πn

両辺に

\frac{π}{8}

を足す

a - \frac{π}{8} + \frac{π}{8} = π - arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

簡素化

a = π - arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

a = π - arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

a = arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}, a = π - arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

sin (a - \frac{π}{8}) = - \frac{2}{3} : a = - arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}, a = π + arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

sin (a - \frac{π}{8}) = - \frac{2}{3}

三角関数の逆数プロパティを適用する

sin (a - \frac{π}{8}) = - \frac{2}{3}

以下の一般解

sin (a - \frac{π}{8}) = - \frac{2}{3}

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

a - \frac{π}{8} = arcsin (- \frac{2}{3}) + 2 πn, a - \frac{π}{8} = π + arcsin (\frac{2}{3}) + 2 πn

a - \frac{π}{8} = arcsin (- \frac{2}{3}) + 2 πn, a - \frac{π}{8} = π + arcsin (\frac{2}{3}) + 2 πn

解く

a - \frac{π}{8} = arcsin (- \frac{2}{3}) + 2 πn : a = - arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

a - \frac{π}{8} = arcsin (- \frac{2}{3}) + 2 πn

簡素化

arcsin (- \frac{2}{3}) + 2 πn : - arcsin (\frac{2}{3}) + 2 πn

arcsin (- \frac{2}{3}) + 2 πn

次のプロパティを使用する:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{2}{3}) = - arcsin (\frac{2}{3})

= - arcsin (\frac{2}{3}) + 2 πn

a - \frac{π}{8} = - arcsin (\frac{2}{3}) + 2 πn

\frac{π}{8}

を右側に移動します

a - \frac{π}{8} = - arcsin (\frac{2}{3}) + 2 πn

両辺に

\frac{π}{8}

を足す

a - \frac{π}{8} + \frac{π}{8} = - arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

簡素化

a = - arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

a = - arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

解く

a - \frac{π}{8} = π + arcsin (\frac{2}{3}) + 2 πn : a = π + arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

a - \frac{π}{8} = π + arcsin (\frac{2}{3}) + 2 πn

\frac{π}{8}

を右側に移動します

a - \frac{π}{8} = π + arcsin (\frac{2}{3}) + 2 πn

両辺に

\frac{π}{8}

を足す

a - \frac{π}{8} + \frac{π}{8} = π + arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

簡素化

a = π + arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

a = π + arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

a = - arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}, a = π + arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

すべての解を組み合わせる

a = arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}, a = π - arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}, a = - arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}, a = π + arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

10進法形式で解を証明する

a = 0.95531 \dots + 2 πn + \frac{π}{8}, a = π - 0.95531 \dots + 2 πn + \frac{π}{8}, a = - 0.95531 \dots + 2 πn + \frac{π}{8}, a = π + 0.95531 \dots + 2 πn + \frac{π}{8}

グラフ

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