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人気のある三角関数 >

20sin^2(x)-21sin(x)+4=0

解

20 sin^{2} (x) - 21 sin (x) + 4 = 0

解

x = 0.92729 \dots + 2 πn, x = π - 0.92729 \dots + 2 πn, x = 0.25268 \dots + 2 πn, x = π - 0.25268 \dots + 2 πn

+1

度

x = 53.13010 \dots^{\circ} + 36 0^{\circ} n, x = 126.86989 \dots^{\circ} + 36 0^{\circ} n, x = 14.47751 \dots^{\circ} + 36 0^{\circ} n, x = 165.52248 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

20 sin^{2} (x) - 21 sin (x) + 4 = 0

置換で解く

20 sin^{2} (x) - 21 sin (x) + 4 = 0

仮定:

sin (x) = u

20 u^{2} - 21 u + 4 = 0

20 u^{2} - 21 u + 4 = 0 : u = \frac{4}{5}, u = \frac{1}{4}

20 u^{2} - 21 u + 4 = 0

解くとthe二次式

20 u^{2} - 21 u + 4 = 0

二次Equationの公式:

次の場合:

a = 20, b = - 21, c = 4

u_{1, 2} = \frac{- ( - 21 ) \pm ( - 21 ) ^{2} - 4 \cdot 20 \cdot 4}{2 \cdot 20}

u_{1, 2} = \frac{- ( - 21 ) \pm ( - 21 ) ^{2} - 4 \cdot 20 \cdot 4}{2 \cdot 20}

(- 21)^{2} - 4 \cdot 20 \cdot 4 = 11

(- 21)^{2} - 4 \cdot 20 \cdot 4

指数の規則を適用する:

n

が偶数であれば

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

(- 21)^{2} = 2 1^{2}

= 2 1^{2} - 4 \cdot 20 \cdot 4

数を乗じる:

4 \cdot 20 \cdot 4 = 320

= 2 1^{2} - 320

2 1^{2} = 441

= 441 - 320

数を引く:

441 - 320 = 121

= 121

数を因数に分解する:

121 = 1 1^{2}

= 1 1^{2}

累乗根の規則を適用する:

n a^{n} = a

1 1^{2} = 11

= 11

u_{1, 2} = \frac{- ( - 21 ) \pm 11}{2 \cdot 20}

解を分離する

u_{1} = \frac{- ( - 21 ) + 11}{2 \cdot 20}, u_{2} = \frac{- ( - 21 ) - 11}{2 \cdot 20}

u = \frac{- ( - 21 ) + 11}{2 \cdot 20} : \frac{4}{5}

\frac{- ( - 21 ) + 11}{2 \cdot 20}

規則を適用

- (- a) = a

= \frac{21 + 11}{2 \cdot 20}

数を足す:

21 + 11 = 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{- ( - 21 ) - 11}{2 \cdot 20} : \frac{1}{4}

\frac{- ( - 21 ) - 11}{2 \cdot 20}

規則を適用

- (- a) = a

= \frac{21 - 11}{2 \cdot 20}

数を引く:

21 - 11 = 10

= \frac{10}{2 \cdot 20}

数を乗じる:

2 \cdot 20 = 40

= \frac{10}{40}

共通因数を約分する:

10

= \frac{1}{4}

二次equationの解:

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

代用を戻す

u = sin (x)

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

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

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}{4} : x = arcsin (\frac{1}{4}) + 2 πn, x = π - arcsin (\frac{1}{4}) + 2 πn

sin (x) = \frac{1}{4}

三角関数の逆数プロパティを適用する

sin (x) = \frac{1}{4}

以下の一般解

sin (x) = \frac{1}{4}

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

x = arcsin (\frac{1}{4}) + 2 πn, x = π - arcsin (\frac{1}{4}) + 2 πn

x = arcsin (\frac{1}{4}) + 2 πn, x = π - arcsin (\frac{1}{4}) + 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 = arcsin (\frac{1}{4}) + 2 πn, x = π - arcsin (\frac{1}{4}) + 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 = 0.25268 \dots + 2 πn, x = π - 0.25268 \dots + 2 πn

グラフ

人気の例

4 tan (x) + 1 = 0

cot^2(θ)=-cot(θ)

cot^{2} (θ) = - cot (θ)

0.2727=sec(2.746x)-1

0.2727 = sec (2.746 x) - 1

2sqrt(3)sin(2θ)=sqrt(3),0<= θ<2pi

23 sin (2 θ) = 3, 0 \leq θ < 2 π

cos(x)-sqrt(3)=3cos(x)

cos (x) - 3 = 3 cos (x)