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人気のある三角関数 >

sin^2(x)+3sin(x)-1=0

解

sin^{2} (x) + 3 sin (x) - 1 = 0

解

x = 0.30760 \dots + 2 πn, x = π - 0.30760 \dots + 2 πn

+1

度

x = 17.62439 \dots^{\circ} + 36 0^{\circ} n, x = 162.37560 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

sin^{2} (x) + 3 sin (x) - 1 = 0

置換で解く

sin^{2} (x) + 3 sin (x) - 1 = 0

仮定:

sin (x) = u

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

u^{2} + 3 u - 1 = 0 : u = \frac{- 3 + 13}{2}, u = \frac{- 3 - 13}{2}

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

解くとthe二次式

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

二次Equationの公式:

次の場合:

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

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

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

3^{2} - 4 \cdot 1 \cdot (- 1) = 13

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

規則を適用

- (- a) = a

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

数を乗じる:

4 \cdot 1 \cdot 1 = 4

= 3^{2} + 4

3^{2} = 9

= 9 + 4

数を足す:

9 + 4 = 13

= 13

u_{1, 2} = \frac{- 3 \pm 13}{2 \cdot 1}

解を分離する

u_{1} = \frac{- 3 + 13}{2 \cdot 1}, u_{2} = \frac{- 3 - 13}{2 \cdot 1}

u = \frac{- 3 + 13}{2 \cdot 1} : \frac{- 3 + 13}{2}

\frac{- 3 + 13}{2 \cdot 1}

数を乗じる:

2 \cdot 1 = 2

= \frac{- 3 + 13}{2}

u = \frac{- 3 - 13}{2 \cdot 1} : \frac{- 3 - 13}{2}

\frac{- 3 - 13}{2 \cdot 1}

数を乗じる:

2 \cdot 1 = 2

= \frac{- 3 - 13}{2}

二次equationの解:

u = \frac{- 3 + 13}{2}, u = \frac{- 3 - 13}{2}

代用を戻す

u = sin (x)

sin (x) = \frac{- 3 + 13}{2}, sin (x) = \frac{- 3 - 13}{2}

sin (x) = \frac{- 3 + 13}{2}, sin (x) = \frac{- 3 - 13}{2}

sin (x) = \frac{- 3 + 13}{2} : x = arcsin (\frac{- 3 + 13}{2}) + 2 πn, x = π - arcsin (\frac{- 3 + 13}{2}) + 2 πn

sin (x) = \frac{- 3 + 13}{2}

三角関数の逆数プロパティを適用する

sin (x) = \frac{- 3 + 13}{2}

以下の一般解

sin (x) = \frac{- 3 + 13}{2}

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

x = arcsin (\frac{- 3 + 13}{2}) + 2 πn, x = π - arcsin (\frac{- 3 + 13}{2}) + 2 πn

x = arcsin (\frac{- 3 + 13}{2}) + 2 πn, x = π - arcsin (\frac{- 3 + 13}{2}) + 2 πn

sin (x) = \frac{- 3 - 13}{2} :

解なし

sin (x) = \frac{- 3 - 13}{2}

- 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 + 13}{2}) + 2 πn, x = π - arcsin (\frac{- 3 + 13}{2}) + 2 πn

10進法形式で解を証明する

x = 0.30760 \dots + 2 πn, x = π - 0.30760 \dots + 2 πn

グラフ

人気の例

tan(x)-3^{1/2}=0

(1+(2sin(x)))/((cos(x)))=0

(sin^2(a))/2 =(1-cos(a))/2

(cot^2(x))/2 =3