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人気のある三角関数 >

8cos^2(x)+2sin(x)-7=0

解

8 cos^{2} (x) + 2 sin (x) - 7 = 0

解

x = - 0.25268 \dots + 2 πn, x = π + 0.25268 \dots + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

+1

度

x = - 14.47751 \dots^{\circ} + 36 0^{\circ} n, x = 194.47751 \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

8 cos^{2} (x) + 2 sin (x) - 7 = 0

三角関数の公式を使用して書き換える

- 7 + 2 sin (x) + 8 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)

= - 7 + 2 sin (x) + 8 (1 - sin^{2} (x))

簡素化

- 7 + 2 sin (x) + 8 (1 - sin^{2} (x)) : 2 sin (x) - 8 sin^{2} (x) + 1

- 7 + 2 sin (x) + 8 (1 - sin^{2} (x))

拡張

8 (1 - sin^{2} (x)) : 8 - 8 sin^{2} (x)

8 (1 - sin^{2} (x))

分配法則を適用する:

a (b - c) = ab - a c

a = 8, b = 1, c = sin^{2} (x)

= 8 \cdot 1 - 8 sin^{2} (x)

数を乗じる:

8 \cdot 1 = 8

= 8 - 8 sin^{2} (x)

= - 7 + 2 sin (x) + 8 - 8 sin^{2} (x)

簡素化

- 7 + 2 sin (x) + 8 - 8 sin^{2} (x) : 2 sin (x) - 8 sin^{2} (x) + 1

- 7 + 2 sin (x) + 8 - 8 sin^{2} (x)

条件のようなグループ

= 2 sin (x) - 8 sin^{2} (x) - 7 + 8

数を足す/引く:

- 7 + 8 = 1

= 2 sin (x) - 8 sin^{2} (x) + 1

= 2 sin (x) - 8 sin^{2} (x) + 1

= 2 sin (x) - 8 sin^{2} (x) + 1

1 + 2 sin (x) - 8 sin^{2} (x) = 0

置換で解く

1 + 2 sin (x) - 8 sin^{2} (x) = 0

仮定:

sin (x) = u

1 + 2 u - 8 u^{2} = 0

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

1 + 2 u - 8 u^{2} = 0

標準的な形式で書く

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

- 8 u^{2} + 2 u + 1 = 0

解くとthe二次式

- 8 u^{2} + 2 u + 1 = 0

二次Equationの公式:

次の場合:

a = - 8, b = 2, c = 1

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

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

2^{2} - 4 (- 8) \cdot 1 = 6

2^{2} - 4 (- 8) \cdot 1

規則を適用

- (- a) = a

= 2^{2} + 4 \cdot 8 \cdot 1

数を乗じる:

4 \cdot 8 \cdot 1 = 32

= 2^{2} + 32

2^{2} = 4

= 4 + 32

数を足す:

4 + 32 = 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{- 2 \pm 6}{2 ( - 8 )}

解を分離する

u_{1} = \frac{- 2 + 6}{2 ( - 8 )}, u_{2} = \frac{- 2 - 6}{2 ( - 8 )}

u = \frac{- 2 + 6}{2 ( - 8 )} : - \frac{1}{4}

\frac{- 2 + 6}{2 ( - 8 )}

括弧を削除する:

(- a) = - a

= \frac{- 2 + 6}{- 2 \cdot 8}

数を足す/引く:

- 2 + 6 = 4

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

数を乗じる:

2 \cdot 8 = 16

= \frac{4}{- 16}

分数の規則を適用する:

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

= - \frac{4}{16}

共通因数を約分する:

4

= - \frac{1}{4}

u = \frac{- 2 - 6}{2 ( - 8 )} : \frac{1}{2}

\frac{- 2 - 6}{2 ( - 8 )}

括弧を削除する:

(- a) = - a

= \frac{- 2 - 6}{- 2 \cdot 8}

数を引く:

- 2 - 6 = - 8

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

数を乗じる:

2 \cdot 8 = 16

= \frac{- 8}{- 16}

分数の規則を適用する:

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

= \frac{8}{16}

共通因数を約分する:

8

= \frac{1}{2}

二次equationの解:

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

代用を戻す

u = sin (x)

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

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

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

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{1}{4}) + 2 πn, x = π + arcsin (\frac{1}{4}) + 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 = - 0.25268 \dots + 2 πn, x = π + 0.25268 \dots + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

グラフ

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