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人気のある三角関数 >

3sin^2(x)+5cos(x)=1

解

3 sin^{2} (x) + 5 cos (x) = 1

解

x = 1.91063 \dots + 2 πn, x = - 1.91063 \dots + 2 πn

+1

度

x = 109.47122 \dots^{\circ} + 36 0^{\circ} n, x = - 109.47122 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

3 sin^{2} (x) + 5 cos (x) = 1

両辺から

1

を引く

3 sin^{2} (x) + 5 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) + 5 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)) + 5 cos (x)

簡素化

- 1 + 3 (1 - cos^{2} (x)) + 5 cos (x) : 5 cos (x) - 3 cos^{2} (x) + 2

- 1 + 3 (1 - cos^{2} (x)) + 5 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) + 5 cos (x)

数を足す/引く:

- 1 + 3 = 2

= 5 cos (x) - 3 cos^{2} (x) + 2

= 5 cos (x) - 3 cos^{2} (x) + 2

2 - 3 cos^{2} (x) + 5 cos (x) = 0

置換で解く

2 - 3 cos^{2} (x) + 5 cos (x) = 0

仮定:

cos (x) = u

2 - 3 u^{2} + 5 u = 0

2 - 3 u^{2} + 5 u = 0 : u = - \frac{1}{3}, u = 2

2 - 3 u^{2} + 5 u = 0

標準的な形式で書く

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

- 3 u^{2} + 5 u + 2 = 0

解くとthe二次式

- 3 u^{2} + 5 u + 2 = 0

二次Equationの公式:

次の場合:

a = - 3, b = 5, c = 2

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 ( - 3 ) \cdot 2}{2 ( - 3 )}

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 ( - 3 ) \cdot 2}{2 ( - 3 )}

5^{2} - 4 (- 3) \cdot 2 = 7

5^{2} - 4 (- 3) \cdot 2

規則を適用

- (- a) = a

= 5^{2} + 4 \cdot 3 \cdot 2

数を乗じる:

4 \cdot 3 \cdot 2 = 24

= 5^{2} + 24

5^{2} = 25

= 25 + 24

数を足す:

25 + 24 = 49

= 49

数を因数に分解する:

49 = 7^{2}

= 7^{2}

累乗根の規則を適用する:

n a^{n} = a

7^{2} = 7

= 7

u_{1, 2} = \frac{- 5 \pm 7}{2 ( - 3 )}

解を分離する

u_{1} = \frac{- 5 + 7}{2 ( - 3 )}, u_{2} = \frac{- 5 - 7}{2 ( - 3 )}

u = \frac{- 5 + 7}{2 ( - 3 )} : - \frac{1}{3}

\frac{- 5 + 7}{2 ( - 3 )}

括弧を削除する:

(- a) = - a

= \frac{- 5 + 7}{- 2 \cdot 3}

数を足す/引く:

- 5 + 7 = 2

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

数を乗じる:

2 \cdot 3 = 6

= \frac{2}{- 6}

分数の規則を適用する:

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

= - \frac{2}{6}

共通因数を約分する:

2

= - \frac{1}{3}

u = \frac{- 5 - 7}{2 ( - 3 )} : 2

\frac{- 5 - 7}{2 ( - 3 )}

括弧を削除する:

(- a) = - a

= \frac{- 5 - 7}{- 2 \cdot 3}

数を引く:

- 5 - 7 = - 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

二次equationの解:

u = - \frac{1}{3}, u = 2

代用を戻す

u = cos (x)

cos (x) = - \frac{1}{3}, cos (x) = 2

cos (x) = - \frac{1}{3}, cos (x) = 2

cos (x) = - \frac{1}{3} : x = arccos (- \frac{1}{3}) + 2 πn, x = - arccos (- \frac{1}{3}) + 2 πn

cos (x) = - \frac{1}{3}

三角関数の逆数プロパティを適用する

cos (x) = - \frac{1}{3}

以下の一般解

cos (x) = - \frac{1}{3}

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

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

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

cos (x) = 2 :

解なし

cos (x) = 2

- 1 \leq cos (x) \leq 1

解なし

すべての解を組み合わせる

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

10進法形式で解を証明する

x = 1.91063 \dots + 2 πn, x = - 1.91063 \dots + 2 πn

グラフ

人気の例

2sin(x)+cos(x)=-2

2 sin (x) + cos (x) = - 2

sin(x)-sin(2x)=0,x<= 2pi,0

sin (x) - sin (2 x) = 0, x \leq 2 π, 0

3cos^2(x)=sin^2(x)+2

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

cos (x) = \frac{π}{3}

(sec^2(x))/2 =2cos^2(x)

\frac{sec ^{2} ( x )}{2} = 2 cos^{2} (x)