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人気のある三角関数 >

2sin^2(x)+3sin(x)=1

解

2 sin^{2} (x) + 3 sin (x) = 1

解

x = 0.28460 \dots + 2 πn, x = π - 0.28460 \dots + 2 πn

+1

度

x = 16.30654 \dots^{\circ} + 36 0^{\circ} n, x = 163.69345 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

2 sin^{2} (x) + 3 sin (x) = 1

置換で解く

2 sin^{2} (x) + 3 sin (x) = 1

仮定:

sin (x) = u

2 u^{2} + 3 u = 1

2 u^{2} + 3 u = 1 : u = \frac{- 3 + 17}{4}, u = \frac{- 3 - 17}{4}

2 u^{2} + 3 u = 1

1

を左側に移動します

2 u^{2} + 3 u = 1

両辺から

1

を引く

2 u^{2} + 3 u - 1 = 1 - 1

簡素化

2 u^{2} + 3 u - 1 = 0

2 u^{2} + 3 u - 1 = 0

解くとthe二次式

2 u^{2} + 3 u - 1 = 0

二次Equationの公式:

次の場合:

a = 2, b = 3, c = - 1

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 \cdot 2 ( - 1 )}{2 \cdot 2}

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 \cdot 2 ( - 1 )}{2 \cdot 2}

3^{2} - 4 \cdot 2 (- 1) = 17

3^{2} - 4 \cdot 2 (- 1)

規則を適用

- (- a) = a

= 3^{2} + 4 \cdot 2 \cdot 1

数を乗じる:

4 \cdot 2 \cdot 1 = 8

= 3^{2} + 8

3^{2} = 9

= 9 + 8

数を足す:

9 + 8 = 17

= 17

u_{1, 2} = \frac{- 3 \pm 17}{2 \cdot 2}

解を分離する

u_{1} = \frac{- 3 + 17}{2 \cdot 2}, u_{2} = \frac{- 3 - 17}{2 \cdot 2}

u = \frac{- 3 + 17}{2 \cdot 2} : \frac{- 3 + 17}{4}

\frac{- 3 + 17}{2 \cdot 2}

数を乗じる:

2 \cdot 2 = 4

= \frac{- 3 + 17}{4}

u = \frac{- 3 - 17}{2 \cdot 2} : \frac{- 3 - 17}{4}

\frac{- 3 - 17}{2 \cdot 2}

数を乗じる:

2 \cdot 2 = 4

= \frac{- 3 - 17}{4}

二次equationの解:

u = \frac{- 3 + 17}{4}, u = \frac{- 3 - 17}{4}

代用を戻す

u = sin (x)

sin (x) = \frac{- 3 + 17}{4}, sin (x) = \frac{- 3 - 17}{4}

sin (x) = \frac{- 3 + 17}{4}, sin (x) = \frac{- 3 - 17}{4}

sin (x) = \frac{- 3 + 17}{4} : x = arcsin (\frac{- 3 + 17}{4}) + 2 πn, x = π - arcsin (\frac{- 3 + 17}{4}) + 2 πn

sin (x) = \frac{- 3 + 17}{4}

三角関数の逆数プロパティを適用する

sin (x) = \frac{- 3 + 17}{4}

以下の一般解

sin (x) = \frac{- 3 + 17}{4}

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

x = arcsin (\frac{- 3 + 17}{4}) + 2 πn, x = π - arcsin (\frac{- 3 + 17}{4}) + 2 πn

x = arcsin (\frac{- 3 + 17}{4}) + 2 πn, x = π - arcsin (\frac{- 3 + 17}{4}) + 2 πn

sin (x) = \frac{- 3 - 17}{4} :

解なし

sin (x) = \frac{- 3 - 17}{4}

- 1 \leq sin (x) \leq 1

解なし

すべての解を組み合わせる

x = arcsin (\frac{- 3 + 17}{4}) + 2 πn, x = π - arcsin (\frac{- 3 + 17}{4}) + 2 πn

10進法形式で解を証明する

x = 0.28460 \dots + 2 πn, x = π - 0.28460 \dots + 2 πn

グラフ

人気の例

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11.33 = 1.59 cos (0.9 9^{\circ} (x - 18 2^{\circ})) + 12.14

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4cos(x)=-2sqrt(2)

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