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人気のある三角関数 >

(3cos(x)-6sin(x))^2-27sin^2(x)=-9

解

(3 cos (x) - 6 sin (x))^{2} - 27 sin^{2} (x) = - 9

解

x = \frac{π}{4} + πn

+1

度

x = 4 5^{\circ} + 18 0^{\circ} n

解答ステップ

(3 cos (x) - 6 sin (x))^{2} - 27 sin^{2} (x) = - 9

両辺から

- 9

を引く

9 cos^{2} (x) - 36 cos (x) sin (x) + 9 sin^{2} (x) + 9 = 0

三角関数の公式を使用して書き換える

9 + 9 cos^{2} (x) + 9 sin^{2} (x) - 36 cos (x) sin (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 + 9 (1 - sin^{2} (x)) + 9 sin^{2} (x) - 36 cos (x) sin (x)

簡素化

9 + 9 (1 - sin^{2} (x)) + 9 sin^{2} (x) - 36 cos (x) sin (x) : - 36 cos (x) sin (x) + 18

9 + 9 (1 - sin^{2} (x)) + 9 sin^{2} (x) - 36 cos (x) sin (x)

拡張

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

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

分配法則を適用する:

a (b - c) = ab - a c

a = 9, b = 1, c = sin^{2} (x)

= 9 \cdot 1 - 9 sin^{2} (x)

数を乗じる:

9 \cdot 1 = 9

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

= 9 + 9 - 9 sin^{2} (x) + 9 sin^{2} (x) - 36 cos (x) sin (x)

簡素化

9 + 9 - 9 sin^{2} (x) + 9 sin^{2} (x) - 36 cos (x) sin (x) : - 36 cos (x) sin (x) + 18

9 + 9 - 9 sin^{2} (x) + 9 sin^{2} (x) - 36 cos (x) sin (x)

類似した元を足す:

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

= 9 + 9 - 36 cos (x) sin (x)

数を足す:

9 + 9 = 18

= - 36 cos (x) sin (x) + 18

= - 36 cos (x) sin (x) + 18

= - 36 cos (x) sin (x) + 18

2倍角の公式を使用:

2 sin (x) cos (x) = sin (2 x)

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

= 18 - 36 \cdot \frac{sin ( 2 x )}{2}

18 - 36 \cdot \frac{sin ( 2 x )}{2} = 0

36 \cdot \frac{sin ( 2 x )}{2} = 18 sin (2 x)

36 \cdot \frac{sin ( 2 x )}{2}

分数を乗じる:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{sin ( 2 x ) \cdot 36}{2}

数を割る:

\frac{36}{2} = 18

= 18 sin (2 x)

18 - 18 sin (2 x) = 0

18

を右側に移動します

18 - 18 sin (2 x) = 0

両辺から

18

を引く

18 - 18 sin (2 x) - 18 = 0 - 18

簡素化

- 18 sin (2 x) = - 18

- 18 sin (2 x) = - 18

以下で両辺を割る

- 18

- 18 sin (2 x) = - 18

以下で両辺を割る

- 18

\frac{- 18 sin ( 2 x )}{- 18} = \frac{- 18}{- 18}

簡素化

sin (2 x) = 1

sin (2 x) = 1

以下の一般解

sin (2 x) = 1

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}

2 x = \frac{π}{2} + 2 πn

2 x = \frac{π}{2} + 2 πn

解く

2 x = \frac{π}{2} + 2 πn : x = \frac{π}{4} + πn

2 x = \frac{π}{2} + 2 πn

以下で両辺を割る

2

2 x = \frac{π}{2} + 2 πn

以下で両辺を割る

2

\frac{2 x}{2} = \frac{\frac{π}{2}}{2} + \frac{2 πn}{2}

簡素化

\frac{2 x}{2} = \frac{\frac{π}{2}}{2} + \frac{2 πn}{2}

簡素化

\frac{2 x}{2} : x

\frac{2 x}{2}

数を割る:

\frac{2}{2} = 1

= x

簡素化

\frac{\frac{π}{2}}{2} + \frac{2 πn}{2} : \frac{π}{4} + πn

\frac{\frac{π}{2}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{2}}{2} = \frac{π}{4}

\frac{\frac{π}{2}}{2}

分数の規則を適用する:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{π}{2 \cdot 2}

数を乗じる:

2 \cdot 2 = 4

= \frac{π}{4}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数を割る:

\frac{2}{2} = 1

= πn

= \frac{π}{4} + πn

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

グラフ

人気の例

3(cos(x)-sin(x))=0

3 (cos (x) - sin (x)) = 0

4tan(x)+4sqrt(3)=0

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