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人気のある三角関数 >

12sin^2(x)-17sin(x)+6=0

解

12 sin^{2} (x) - 17 sin (x) + 6 = 0

解

x = 0.84806 \dots + 2 πn, x = π - 0.84806 \dots + 2 πn, x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

+1

度

x = 48.59037 \dots^{\circ} + 36 0^{\circ} n, x = 131.40962 \dots^{\circ} + 36 0^{\circ} n, x = 41.81031 \dots^{\circ} + 36 0^{\circ} n, x = 138.18968 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

12 sin^{2} (x) - 17 sin (x) + 6 = 0

置換で解く

12 sin^{2} (x) - 17 sin (x) + 6 = 0

仮定:

sin (x) = u

12 u^{2} - 17 u + 6 = 0

12 u^{2} - 17 u + 6 = 0 : u = \frac{3}{4}, u = \frac{2}{3}

12 u^{2} - 17 u + 6 = 0

解くとthe二次式

12 u^{2} - 17 u + 6 = 0

二次Equationの公式:

次の場合:

a = 12, b = - 17, c = 6

u_{1, 2} = \frac{- ( - 17 ) \pm ( - 17 ) ^{2} - 4 \cdot 12 \cdot 6}{2 \cdot 12}

u_{1, 2} = \frac{- ( - 17 ) \pm ( - 17 ) ^{2} - 4 \cdot 12 \cdot 6}{2 \cdot 12}

(- 17)^{2} - 4 \cdot 12 \cdot 6 = 1

(- 17)^{2} - 4 \cdot 12 \cdot 6

指数の規則を適用する:

n

が偶数であれば

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

(- 17)^{2} = 1 7^{2}

= 1 7^{2} - 4 \cdot 12 \cdot 6

数を乗じる:

4 \cdot 12 \cdot 6 = 288

= 1 7^{2} - 288

1 7^{2} = 289

= 289 - 288

数を引く:

289 - 288 = 1

= 1

規則を適用

1 = 1

= 1

u_{1, 2} = \frac{- ( - 17 ) \pm 1}{2 \cdot 12}

解を分離する

u_{1} = \frac{- ( - 17 ) + 1}{2 \cdot 12}, u_{2} = \frac{- ( - 17 ) - 1}{2 \cdot 12}

u = \frac{- ( - 17 ) + 1}{2 \cdot 12} : \frac{3}{4}

\frac{- ( - 17 ) + 1}{2 \cdot 12}

規則を適用

- (- a) = a

= \frac{17 + 1}{2 \cdot 12}

数を足す:

17 + 1 = 18

= \frac{18}{2 \cdot 12}

数を乗じる:

2 \cdot 12 = 24

= \frac{18}{24}

共通因数を約分する:

6

= \frac{3}{4}

u = \frac{- ( - 17 ) - 1}{2 \cdot 12} : \frac{2}{3}

\frac{- ( - 17 ) - 1}{2 \cdot 12}

規則を適用

- (- a) = a

= \frac{17 - 1}{2 \cdot 12}

数を引く:

17 - 1 = 16

= \frac{16}{2 \cdot 12}

数を乗じる:

2 \cdot 12 = 24

= \frac{16}{24}

共通因数を約分する:

8

= \frac{2}{3}

二次equationの解:

u = \frac{3}{4}, u = \frac{2}{3}

代用を戻す

u = sin (x)

sin (x) = \frac{3}{4}, sin (x) = \frac{2}{3}

sin (x) = \frac{3}{4}, sin (x) = \frac{2}{3}

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

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

三角関数の逆数プロパティを適用する

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

以下の一般解

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

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

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

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

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

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

三角関数の逆数プロパティを適用する

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

以下の一般解

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

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

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

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

すべての解を組み合わせる

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

10進法形式で解を証明する

x = 0.84806 \dots + 2 πn, x = π - 0.84806 \dots + 2 πn, x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

グラフ

人気の例

sin(θ)=(sqrt(3))/3

sin (θ) = \frac{3}{3}

tan(x+pi)+cos(x+pi/2)=0

tan (x + π) + cos (x + \frac{π}{2}) = 0

sin(θ)cos(θ)=(sqrt(3))/4

sin (θ) cos (θ) = \frac{3}{4}

tan(x+pi/6)=sqrt(3)

tan (x + \frac{π}{6}) = 3

cos(2θ)-sin^2(θ)=0

cos (2 θ) - sin^{2} (θ) = 0