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人気のある三角関数 >

9sin^2(x)-6sin(x)+1=0

解

9 sin^{2} (x) - 6 sin (x) + 1 = 0

解

x = 0.33983 \dots + 2 πn, x = π - 0.33983 \dots + 2 πn

+1

度

x = 19.47122 \dots^{\circ} + 36 0^{\circ} n, x = 160.52877 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

9 sin^{2} (x) - 6 sin (x) + 1 = 0

置換で解く

9 sin^{2} (x) - 6 sin (x) + 1 = 0

仮定:

sin (x) = u

9 u^{2} - 6 u + 1 = 0

9 u^{2} - 6 u + 1 = 0 : u = \frac{1}{3}

9 u^{2} - 6 u + 1 = 0

解くとthe二次式

9 u^{2} - 6 u + 1 = 0

二次Equationの公式:

次の場合:

a = 9, b = - 6, c = 1

u_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 \cdot 9 \cdot 1}{2 \cdot 9}

u_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 \cdot 9 \cdot 1}{2 \cdot 9}

(- 6)^{2} - 4 \cdot 9 \cdot 1 = 0

(- 6)^{2} - 4 \cdot 9 \cdot 1

指数の規則を適用する:

n

が偶数であれば

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

(- 6)^{2} = 6^{2}

= 6^{2} - 4 \cdot 9 \cdot 1

数を乗じる:

4 \cdot 9 \cdot 1 = 36

= 6^{2} - 36

6^{2} = 36

= 36 - 36

数を引く:

36 - 36 = 0

= 0

u_{1, 2} = \frac{- ( - 6 ) \pm 0}{2 \cdot 9}

u = \frac{- ( - 6 )}{2 \cdot 9}

\frac{- ( - 6 )}{2 \cdot 9} = \frac{1}{3}

\frac{- ( - 6 )}{2 \cdot 9}

規則を適用

- (- a) = a

= \frac{6}{2 \cdot 9}

数を乗じる:

2 \cdot 9 = 18

= \frac{6}{18}

共通因数を約分する:

6

= \frac{1}{3}

u = \frac{1}{3}

二次equationの解:

u = \frac{1}{3}

代用を戻す

u = sin (x)

sin (x) = \frac{1}{3}

sin (x) = \frac{1}{3}

sin (x) = \frac{1}{3} : x = arcsin (\frac{1}{3}) + 2 πn, x = π - arcsin (\frac{1}{3}) + 2 πn

sin (x) = \frac{1}{3}

三角関数の逆数プロパティを適用する

sin (x) = \frac{1}{3}

以下の一般解

sin (x) = \frac{1}{3}

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

x = arcsin (\frac{1}{3}) + 2 πn, x = π - arcsin (\frac{1}{3}) + 2 πn

x = arcsin (\frac{1}{3}) + 2 πn, x = π - arcsin (\frac{1}{3}) + 2 πn

すべての解を組み合わせる

x = arcsin (\frac{1}{3}) + 2 πn, x = π - arcsin (\frac{1}{3}) + 2 πn

10進法形式で解を証明する

x = 0.33983 \dots + 2 πn, x = π - 0.33983 \dots + 2 πn

グラフ

人気の例

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cos (a) + 1 = 4 cos (a) + 1

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