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人気のある三角関数 >

3sin^2(x)-4cos(x)+1=0

解

3 sin^{2} (x) - 4 cos (x) + 1 = 0

解

x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn

+1

度

x = 48.18968 \dots^{\circ} + 36 0^{\circ} n, x = 311.81031 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

3 sin^{2} (x) - 4 cos (x) + 1 = 0

三角関数の公式を使用して書き換える

1 + 3 sin^{2} (x) - 4 cos (x)

ピタゴラスの公式を使用する:

cos^{2} (x) + sin^{2} (x) = 1

sin^{2} (x) = 1 - cos^{2} (x)

= 1 + 3 (1 - cos^{2} (x)) - 4 cos (x)

簡素化

1 + 3 (1 - cos^{2} (x)) - 4 cos (x) : - 3 cos^{2} (x) - 4 cos (x) + 4

1 + 3 (1 - cos^{2} (x)) - 4 cos (x)

拡張

3 (1 - cos^{2} (x)) : 3 - 3 cos^{2} (x)

3 (1 - cos^{2} (x))

分配法則を適用する:

a (b - c) = ab - a c

a = 3, b = 1, c = cos^{2} (x)

= 3 \cdot 1 - 3 cos^{2} (x)

数を乗じる:

3 \cdot 1 = 3

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

= 1 + 3 - 3 cos^{2} (x) - 4 cos (x)

数を足す:

1 + 3 = 4

= - 3 cos^{2} (x) - 4 cos (x) + 4

= - 3 cos^{2} (x) - 4 cos (x) + 4

4 - 3 cos^{2} (x) - 4 cos (x) = 0

置換で解く

4 - 3 cos^{2} (x) - 4 cos (x) = 0

仮定:

cos (x) = u

4 - 3 u^{2} - 4 u = 0

4 - 3 u^{2} - 4 u = 0 : u = - 2, u = \frac{2}{3}

4 - 3 u^{2} - 4 u = 0

標準的な形式で書く

a x^{2} + b x + c = 0

- 3 u^{2} - 4 u + 4 = 0

解くとthe二次式

- 3 u^{2} - 4 u + 4 = 0

二次Equationの公式:

次の場合:

a = - 3, b = - 4, c = 4

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

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

(- 4)^{2} - 4 (- 3) \cdot 4 = 8

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

規則を適用

- (- a) = a

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

指数の規則を適用する:

n

が偶数であれば

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

(- 4)^{2} = 4^{2}

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

数を乗じる:

4 \cdot 3 \cdot 4 = 48

= 4^{2} + 48

4^{2} = 16

= 16 + 48

数を足す:

16 + 48 = 64

= 64

数を因数に分解する:

64 = 8^{2}

= 8^{2}

累乗根の規則を適用する:

n a^{n} = a

8^{2} = 8

= 8

u_{1, 2} = \frac{- ( - 4 ) \pm 8}{2 ( - 3 )}

解を分離する

u_{1} = \frac{- ( - 4 ) + 8}{2 ( - 3 )}, u_{2} = \frac{- ( - 4 ) - 8}{2 ( - 3 )}

u = \frac{- ( - 4 ) + 8}{2 ( - 3 )} : - 2

\frac{- ( - 4 ) + 8}{2 ( - 3 )}

括弧を削除する:

(- a) = - a, - (- a) = a

= \frac{4 + 8}{- 2 \cdot 3}

数を足す:

4 + 8 = 12

= \frac{12}{- 2 \cdot 3}

数を乗じる:

2 \cdot 3 = 6

= \frac{12}{- 6}

分数の規則を適用する:

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

= - \frac{12}{6}

数を割る:

\frac{12}{6} = 2

= - 2

u = \frac{- ( - 4 ) - 8}{2 ( - 3 )} : \frac{2}{3}

\frac{- ( - 4 ) - 8}{2 ( - 3 )}

括弧を削除する:

(- a) = - a, - (- a) = a

= \frac{4 - 8}{- 2 \cdot 3}

数を引く:

4 - 8 = - 4

= \frac{- 4}{- 2 \cdot 3}

数を乗じる:

2 \cdot 3 = 6

= \frac{- 4}{- 6}

分数の規則を適用する:

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

= \frac{4}{6}

共通因数を約分する:

2

= \frac{2}{3}

二次equationの解:

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

代用を戻す

u = cos (x)

cos (x) = - 2, cos (x) = \frac{2}{3}

cos (x) = - 2, cos (x) = \frac{2}{3}

cos (x) = - 2 :

解なし

cos (x) = - 2

- 1 \leq cos (x) \leq 1

解なし

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

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

三角関数の逆数プロパティを適用する

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

以下の一般解

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

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

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

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

すべての解を組み合わせる

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

10進法形式で解を証明する

x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn

グラフ

人気の例

sin(x)-cos(2x)-1=0

sin (x) - cos (2 x) - 1 = 0

sin^2(x)=sqrt(3)

sin^{2} (x) = 3

solvefor x,c^2=a^2+b^2-2ab*cos(x)

so l v e f or x, c^{2} = a^{2} + b^{2} - 2 ab \cdot cos (x)

sin(-pi/8 c)=-14/14

sin (- \frac{π}{8} c) = - \frac{14}{14}

(25)/(sin(x))=(18)/(sin(36))

\frac{25}{sin ( x )} = \frac{18}{sin ( 3 6 ^{\circ} )}