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人気のある三角関数 >

10sin^2(x)-3sin(x)-1=0

解

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

解

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = - 0.20135 \dots + 2 πn, x = π + 0.20135 \dots + 2 πn

+1

度

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = - 11.53695 \dots^{\circ} + 36 0^{\circ} n, x = 191.53695 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

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

置換で解く

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

仮定:

sin (x) = u

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

10 u^{2} - 3 u - 1 = 0 : u = \frac{1}{2}, u = - \frac{1}{5}

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

解くとthe二次式

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

二次Equationの公式:

次の場合:

a = 10, b = - 3, c = - 1

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 10 ( - 1 )}{2 \cdot 10}

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 10 ( - 1 )}{2 \cdot 10}

(- 3)^{2} - 4 \cdot 10 (- 1) = 7

(- 3)^{2} - 4 \cdot 10 (- 1)

規則を適用

- (- a) = a

= (- 3)^{2} + 4 \cdot 10 \cdot 1

指数の規則を適用する:

n

が偶数であれば

(- a)^{n} = a^{n}

(- 3)^{2} = 3^{2}

= 3^{2} + 4 \cdot 10 \cdot 1

数を乗じる:

4 \cdot 10 \cdot 1 = 40

= 3^{2} + 40

3^{2} = 9

= 9 + 40

数を足す:

9 + 40 = 49

= 49

数を因数に分解する:

49 = 7^{2}

= 7^{2}

累乗根の規則を適用する:

n a^{n} = a

7^{2} = 7

= 7

u_{1, 2} = \frac{- ( - 3 ) \pm 7}{2 \cdot 10}

解を分離する

u_{1} = \frac{- ( - 3 ) + 7}{2 \cdot 10}, u_{2} = \frac{- ( - 3 ) - 7}{2 \cdot 10}

u = \frac{- ( - 3 ) + 7}{2 \cdot 10} : \frac{1}{2}

\frac{- ( - 3 ) + 7}{2 \cdot 10}

規則を適用

- (- a) = a

= \frac{3 + 7}{2 \cdot 10}

数を足す:

3 + 7 = 10

= \frac{10}{2 \cdot 10}

数を乗じる:

2 \cdot 10 = 20

= \frac{10}{20}

共通因数を約分する:

10

= \frac{1}{2}

u = \frac{- ( - 3 ) - 7}{2 \cdot 10} : - \frac{1}{5}

\frac{- ( - 3 ) - 7}{2 \cdot 10}

規則を適用

- (- a) = a

= \frac{3 - 7}{2 \cdot 10}

数を引く:

3 - 7 = - 4

= \frac{- 4}{2 \cdot 10}

数を乗じる:

2 \cdot 10 = 20

= \frac{- 4}{20}

分数の規則を適用する:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{4}{20}

共通因数を約分する:

4

= - \frac{1}{5}

二次equationの解:

u = \frac{1}{2}, u = - \frac{1}{5}

代用を戻す

u = sin (x)

sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{5}

sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{5}

sin (x) = \frac{1}{2} : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

sin (x) = \frac{1}{2}

以下の一般解

sin (x) = \frac{1}{2}

sin (x)

2 πn

循環を含む周期性テーブル:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

sin (x) = - \frac{1}{5} : x = arcsin (- \frac{1}{5}) + 2 πn, x = π + arcsin (\frac{1}{5}) + 2 πn

sin (x) = - \frac{1}{5}

三角関数の逆数プロパティを適用する

sin (x) = - \frac{1}{5}

以下の一般解

sin (x) = - \frac{1}{5}

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

x = arcsin (- \frac{1}{5}) + 2 πn, x = π + arcsin (\frac{1}{5}) + 2 πn

x = arcsin (- \frac{1}{5}) + 2 πn, x = π + arcsin (\frac{1}{5}) + 2 πn

すべての解を組み合わせる

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = arcsin (- \frac{1}{5}) + 2 πn, x = π + arcsin (\frac{1}{5}) + 2 πn

10進法形式で解を証明する

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = - 0.20135 \dots + 2 πn, x = π + 0.20135 \dots + 2 πn

グラフ

人気の例

cos(2x)-cos(pi/2+x)=1

cos (2 x) - cos (\frac{π}{2} + x) = 1

cos(3x)-cos(7x)=0

cos (3 x) - cos (7 x) = 0

2 sin (x - 1) = 0

tan (x) = 0.42

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

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