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人気のある三角関数 >

21+18cos(x)=16sin^2(x)

解

21 + 18 cos (x) = 16 sin^{2} (x)

解

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = 2.24592 \dots + 2 πn, x = - 2.24592 \dots + 2 πn

+1

度

x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n, x = 128.68218 \dots^{\circ} + 36 0^{\circ} n, x = - 128.68218 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

21 + 18 cos (x) = 16 sin^{2} (x)

両辺から

16 sin^{2} (x)

を引く

21 + 18 cos (x) - 16 sin^{2} (x) = 0

三角関数の公式を使用して書き換える

21 - 16 sin^{2} (x) + 18 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)

= 21 - 16 (1 - cos^{2} (x)) + 18 cos (x)

簡素化

21 - 16 (1 - cos^{2} (x)) + 18 cos (x) : 16 cos^{2} (x) + 18 cos (x) + 5

21 - 16 (1 - cos^{2} (x)) + 18 cos (x)

拡張

- 16 (1 - cos^{2} (x)) : - 16 + 16 cos^{2} (x)

- 16 (1 - cos^{2} (x))

分配法則を適用する:

a (b - c) = ab - a c

a = - 16, b = 1, c = cos^{2} (x)

= - 16 \cdot 1 - (- 16) cos^{2} (x)

マイナス・プラスの規則を適用する

- (- a) = a

= - 16 \cdot 1 + 16 cos^{2} (x)

数を乗じる:

16 \cdot 1 = 16

= - 16 + 16 cos^{2} (x)

= 21 - 16 + 16 cos^{2} (x) + 18 cos (x)

数を引く:

21 - 16 = 5

= 16 cos^{2} (x) + 18 cos (x) + 5

= 16 cos^{2} (x) + 18 cos (x) + 5

5 + 16 cos^{2} (x) + 18 cos (x) = 0

置換で解く

5 + 16 cos^{2} (x) + 18 cos (x) = 0

仮定:

cos (x) = u

5 + 16 u^{2} + 18 u = 0

5 + 16 u^{2} + 18 u = 0 : u = - \frac{1}{2}, u = - \frac{5}{8}

5 + 16 u^{2} + 18 u = 0

標準的な形式で書く

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

16 u^{2} + 18 u + 5 = 0

解くとthe二次式

16 u^{2} + 18 u + 5 = 0

二次Equationの公式:

次の場合:

a = 16, b = 18, c = 5

u_{1, 2} = \frac{- 18 \pm 1 8 ^{2} - 4 \cdot 16 \cdot 5}{2 \cdot 16}

u_{1, 2} = \frac{- 18 \pm 1 8 ^{2} - 4 \cdot 16 \cdot 5}{2 \cdot 16}

1 8^{2} - 4 \cdot 16 \cdot 5 = 2

1 8^{2} - 4 \cdot 16 \cdot 5

数を乗じる:

4 \cdot 16 \cdot 5 = 320

= 1 8^{2} - 320

1 8^{2} = 324

= 324 - 320

数を引く:

324 - 320 = 4

= 4

数を因数に分解する:

4 = 2^{2}

= 2^{2}

累乗根の規則を適用する:

n a^{n} = a

2^{2} = 2

= 2

u_{1, 2} = \frac{- 18 \pm 2}{2 \cdot 16}

解を分離する

u_{1} = \frac{- 18 + 2}{2 \cdot 16}, u_{2} = \frac{- 18 - 2}{2 \cdot 16}

u = \frac{- 18 + 2}{2 \cdot 16} : - \frac{1}{2}

\frac{- 18 + 2}{2 \cdot 16}

数を足す/引く:

- 18 + 2 = - 16

= \frac{- 16}{2 \cdot 16}

数を乗じる:

2 \cdot 16 = 32

= \frac{- 16}{32}

分数の規則を適用する:

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

= - \frac{16}{32}

共通因数を約分する:

16

= - \frac{1}{2}

u = \frac{- 18 - 2}{2 \cdot 16} : - \frac{5}{8}

\frac{- 18 - 2}{2 \cdot 16}

数を引く:

- 18 - 2 = - 20

= \frac{- 20}{2 \cdot 16}

数を乗じる:

2 \cdot 16 = 32

= \frac{- 20}{32}

分数の規則を適用する:

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

= - \frac{20}{32}

共通因数を約分する:

4

= - \frac{5}{8}

二次equationの解:

u = - \frac{1}{2}, u = - \frac{5}{8}

代用を戻す

u = cos (x)

cos (x) = - \frac{1}{2}, cos (x) = - \frac{5}{8}

cos (x) = - \frac{1}{2}, cos (x) = - \frac{5}{8}

cos (x) = - \frac{1}{2} : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

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

以下の一般解

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

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

cos (x) = - \frac{5}{8} : x = arccos (- \frac{5}{8}) + 2 πn, x = - arccos (- \frac{5}{8}) + 2 πn

cos (x) = - \frac{5}{8}

三角関数の逆数プロパティを適用する

cos (x) = - \frac{5}{8}

以下の一般解

cos (x) = - \frac{5}{8}

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

x = arccos (- \frac{5}{8}) + 2 πn, x = - arccos (- \frac{5}{8}) + 2 πn

x = arccos (- \frac{5}{8}) + 2 πn, x = - arccos (- \frac{5}{8}) + 2 πn

すべての解を組み合わせる

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = arccos (- \frac{5}{8}) + 2 πn, x = - arccos (- \frac{5}{8}) + 2 πn

10進法形式で解を証明する

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = 2.24592 \dots + 2 πn, x = - 2.24592 \dots + 2 πn

グラフ

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