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人気のある三角関数 >

20sin^2(x)-26sin(x)+8=0

解

20 sin^{2} (x) - 26 sin (x) + 8 = 0

解

x = 0.92729 \dots + 2 πn, x = π - 0.92729 \dots + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

+1

度

x = 53.13010 \dots^{\circ} + 36 0^{\circ} n, x = 126.86989 \dots^{\circ} + 36 0^{\circ} n, x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

解答ステップ

20 sin^{2} (x) - 26 sin (x) + 8 = 0

置換で解く

20 sin^{2} (x) - 26 sin (x) + 8 = 0

仮定:

sin (x) = u

20 u^{2} - 26 u + 8 = 0

20 u^{2} - 26 u + 8 = 0 : u = \frac{4}{5}, u = \frac{1}{2}

20 u^{2} - 26 u + 8 = 0

解くとthe二次式

20 u^{2} - 26 u + 8 = 0

二次Equationの公式:

次の場合:

a = 20, b = - 26, c = 8

u_{1, 2} = \frac{- ( - 26 ) \pm ( - 26 ) ^{2} - 4 \cdot 20 \cdot 8}{2 \cdot 20}

u_{1, 2} = \frac{- ( - 26 ) \pm ( - 26 ) ^{2} - 4 \cdot 20 \cdot 8}{2 \cdot 20}

(- 26)^{2} - 4 \cdot 20 \cdot 8 = 6

(- 26)^{2} - 4 \cdot 20 \cdot 8

指数の規則を適用する:

n

が偶数であれば

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

(- 26)^{2} = 2 6^{2}

= 2 6^{2} - 4 \cdot 20 \cdot 8

数を乗じる:

4 \cdot 20 \cdot 8 = 640

= 2 6^{2} - 640

2 6^{2} = 676

= 676 - 640

数を引く:

676 - 640 = 36

= 36

数を因数に分解する:

36 = 6^{2}

= 6^{2}

累乗根の規則を適用する:

n a^{n} = a

6^{2} = 6

= 6

u_{1, 2} = \frac{- ( - 26 ) \pm 6}{2 \cdot 20}

解を分離する

u_{1} = \frac{- ( - 26 ) + 6}{2 \cdot 20}, u_{2} = \frac{- ( - 26 ) - 6}{2 \cdot 20}

u = \frac{- ( - 26 ) + 6}{2 \cdot 20} : \frac{4}{5}

\frac{- ( - 26 ) + 6}{2 \cdot 20}

規則を適用

- (- a) = a

= \frac{26 + 6}{2 \cdot 20}

数を足す:

26 + 6 = 32

= \frac{32}{2 \cdot 20}

数を乗じる:

2 \cdot 20 = 40

= \frac{32}{40}

共通因数を約分する:

8

= \frac{4}{5}

u = \frac{- ( - 26 ) - 6}{2 \cdot 20} : \frac{1}{2}

\frac{- ( - 26 ) - 6}{2 \cdot 20}

規則を適用

- (- a) = a

= \frac{26 - 6}{2 \cdot 20}

数を引く:

26 - 6 = 20

= \frac{20}{2 \cdot 20}

数を乗じる:

2 \cdot 20 = 40

= \frac{20}{40}

共通因数を約分する:

20

= \frac{1}{2}

二次equationの解:

u = \frac{4}{5}, u = \frac{1}{2}

代用を戻す

u = sin (x)

sin (x) = \frac{4}{5}, sin (x) = \frac{1}{2}

sin (x) = \frac{4}{5}, sin (x) = \frac{1}{2}

sin (x) = \frac{4}{5} : x = arcsin (\frac{4}{5}) + 2 πn, x = π - arcsin (\frac{4}{5}) + 2 πn

sin (x) = \frac{4}{5}

三角関数の逆数プロパティを適用する

sin (x) = \frac{4}{5}

以下の一般解

sin (x) = \frac{4}{5}

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

x = arcsin (\frac{4}{5}) + 2 πn, x = π - arcsin (\frac{4}{5}) + 2 πn

x = arcsin (\frac{4}{5}) + 2 πn, x = π - arcsin (\frac{4}{5}) + 2 πn

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

すべての解を組み合わせる

x = arcsin (\frac{4}{5}) + 2 πn, x = π - arcsin (\frac{4}{5}) + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

10進法形式で解を証明する

x = 0.92729 \dots + 2 πn, x = π - 0.92729 \dots + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

グラフ

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