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人気のある三角関数 >

15cos^2(x)+sin(x)-9=0

解

15 cos^{2} (x) + sin (x) - 9 = 0

解

x = - 0.64350 \dots + 2 πn, x = π + 0.64350 \dots + 2 πn, x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

+1

度

x = - 36.86989 \dots^{\circ} + 36 0^{\circ} n, x = 216.86989 \dots^{\circ} + 36 0^{\circ} n, x = 41.81031 \dots^{\circ} + 36 0^{\circ} n, x = 138.18968 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

15 cos^{2} (x) + sin (x) - 9 = 0

三角関数の公式を使用して書き換える

- 9 + sin (x) + 15 cos^{2} (x)

ピタゴラスの公式を使用する:

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

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

= - 9 + sin (x) + 15 (1 - sin^{2} (x))

簡素化

- 9 + sin (x) + 15 (1 - sin^{2} (x)) : sin (x) - 15 sin^{2} (x) + 6

- 9 + sin (x) + 15 (1 - sin^{2} (x))

拡張

15 (1 - sin^{2} (x)) : 15 - 15 sin^{2} (x)

15 (1 - sin^{2} (x))

分配法則を適用する:

a (b - c) = ab - a c

a = 15, b = 1, c = sin^{2} (x)

= 15 \cdot 1 - 15 sin^{2} (x)

数を乗じる:

15 \cdot 1 = 15

= 15 - 15 sin^{2} (x)

= - 9 + sin (x) + 15 - 15 sin^{2} (x)

簡素化

- 9 + sin (x) + 15 - 15 sin^{2} (x) : sin (x) - 15 sin^{2} (x) + 6

- 9 + sin (x) + 15 - 15 sin^{2} (x)

条件のようなグループ

= sin (x) - 15 sin^{2} (x) - 9 + 15

数を足す/引く:

- 9 + 15 = 6

= sin (x) - 15 sin^{2} (x) + 6

= sin (x) - 15 sin^{2} (x) + 6

= sin (x) - 15 sin^{2} (x) + 6

6 + sin (x) - 15 sin^{2} (x) = 0

置換で解く

6 + sin (x) - 15 sin^{2} (x) = 0

仮定:

sin (x) = u

6 + u - 15 u^{2} = 0

6 + u - 15 u^{2} = 0 : u = - \frac{3}{5}, u = \frac{2}{3}

6 + u - 15 u^{2} = 0

標準的な形式で書く

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

- 15 u^{2} + u + 6 = 0

解くとthe二次式

- 15 u^{2} + u + 6 = 0

二次Equationの公式:

次の場合:

a = - 15, b = 1, c = 6

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 ( - 15 ) \cdot 6}{2 ( - 15 )}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 ( - 15 ) \cdot 6}{2 ( - 15 )}

1^{2} - 4 (- 15) \cdot 6 = 19

1^{2} - 4 (- 15) \cdot 6

規則を適用

1^{a} = 1

1^{2} = 1

= 1 - 4 (- 15) \cdot 6

規則を適用

- (- a) = a

= 1 + 4 \cdot 15 \cdot 6

数を乗じる:

4 \cdot 15 \cdot 6 = 360

= 1 + 360

数を足す:

1 + 360 = 361

= 361

数を因数に分解する:

361 = 1 9^{2}

= 1 9^{2}

累乗根の規則を適用する:

n a^{n} = a

1 9^{2} = 19

= 19

u_{1, 2} = \frac{- 1 \pm 19}{2 ( - 15 )}

解を分離する

u_{1} = \frac{- 1 + 19}{2 ( - 15 )}, u_{2} = \frac{- 1 - 19}{2 ( - 15 )}

u = \frac{- 1 + 19}{2 ( - 15 )} : - \frac{3}{5}

\frac{- 1 + 19}{2 ( - 15 )}

括弧を削除する:

(- a) = - a

= \frac{- 1 + 19}{- 2 \cdot 15}

数を足す/引く:

- 1 + 19 = 18

= \frac{18}{- 2 \cdot 15}

数を乗じる:

2 \cdot 15 = 30

= \frac{18}{- 30}

分数の規則を適用する:

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

= - \frac{18}{30}

共通因数を約分する:

6

= - \frac{3}{5}

u = \frac{- 1 - 19}{2 ( - 15 )} : \frac{2}{3}

\frac{- 1 - 19}{2 ( - 15 )}

括弧を削除する:

(- a) = - a

= \frac{- 1 - 19}{- 2 \cdot 15}

数を引く:

- 1 - 19 = - 20

= \frac{- 20}{- 2 \cdot 15}

数を乗じる:

2 \cdot 15 = 30

= \frac{- 20}{- 30}

分数の規則を適用する:

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

= \frac{20}{30}

共通因数を約分する:

10

= \frac{2}{3}

二次equationの解:

u = - \frac{3}{5}, u = \frac{2}{3}

代用を戻す

u = sin (x)

sin (x) = - \frac{3}{5}, sin (x) = \frac{2}{3}

sin (x) = - \frac{3}{5}, sin (x) = \frac{2}{3}

sin (x) = - \frac{3}{5} : x = arcsin (- \frac{3}{5}) + 2 πn, x = π + arcsin (\frac{3}{5}) + 2 πn

sin (x) = - \frac{3}{5}

三角関数の逆数プロパティを適用する

sin (x) = - \frac{3}{5}

以下の一般解

sin (x) = - \frac{3}{5}

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

x = arcsin (- \frac{3}{5}) + 2 πn, x = π + arcsin (\frac{3}{5}) + 2 πn

x = arcsin (- \frac{3}{5}) + 2 πn, x = π + arcsin (\frac{3}{5}) + 2 πn

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

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

三角関数の逆数プロパティを適用する

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

以下の一般解

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

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

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

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

すべての解を組み合わせる

x = arcsin (- \frac{3}{5}) + 2 πn, x = π + arcsin (\frac{3}{5}) + 2 πn, x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn

10進法形式で解を証明する

x = - 0.64350 \dots + 2 πn, x = π + 0.64350 \dots + 2 πn, x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

グラフ

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