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人気のある三角関数 >

30sin^2(x)=30+15cos(x)

解

30 sin^{2} (x) = 30 + 15 cos (x)

解

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

+1

度

x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n, x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

解答ステップ

30 sin^{2} (x) = 30 + 15 cos (x)

両辺から

30 + 15 cos (x)

を引く

30 sin^{2} (x) - 30 - 15 cos (x) = 0

三角関数の公式を使用して書き換える

- 30 - 15 cos (x) + 30 sin^{2} (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)

= - 30 - 15 cos (x) + 30 (1 - cos^{2} (x))

簡素化

- 30 - 15 cos (x) + 30 (1 - cos^{2} (x)) : - 30 cos^{2} (x) - 15 cos (x)

- 30 - 15 cos (x) + 30 (1 - cos^{2} (x))

拡張

30 (1 - cos^{2} (x)) : 30 - 30 cos^{2} (x)

30 (1 - cos^{2} (x))

分配法則を適用する:

a (b - c) = ab - a c

a = 30, b = 1, c = cos^{2} (x)

= 30 \cdot 1 - 30 cos^{2} (x)

数を乗じる:

30 \cdot 1 = 30

= 30 - 30 cos^{2} (x)

= - 30 - 15 cos (x) + 30 - 30 cos^{2} (x)

簡素化

- 30 - 15 cos (x) + 30 - 30 cos^{2} (x) : - 30 cos^{2} (x) - 15 cos (x)

- 30 - 15 cos (x) + 30 - 30 cos^{2} (x)

条件のようなグループ

= - 15 cos (x) - 30 cos^{2} (x) - 30 + 30

- 30 + 30 = 0

= - 30 cos^{2} (x) - 15 cos (x)

= - 30 cos^{2} (x) - 15 cos (x)

= - 30 cos^{2} (x) - 15 cos (x)

- 15 cos (x) - 30 cos^{2} (x) = 0

置換で解く

- 15 cos (x) - 30 cos^{2} (x) = 0

仮定:

cos (x) = u

- 15 u - 30 u^{2} = 0

- 15 u - 30 u^{2} = 0 : u = - \frac{1}{2}, u = 0

- 15 u - 30 u^{2} = 0

標準的な形式で書く

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

- 30 u^{2} - 15 u = 0

解くとthe二次式

- 30 u^{2} - 15 u = 0

二次Equationの公式:

次の場合:

a = - 30, b = - 15, c = 0

u_{1, 2} = \frac{- ( - 15 ) \pm ( - 15 ) ^{2} - 4 ( - 30 ) \cdot 0}{2 ( - 30 )}

u_{1, 2} = \frac{- ( - 15 ) \pm ( - 15 ) ^{2} - 4 ( - 30 ) \cdot 0}{2 ( - 30 )}

(- 15)^{2} - 4 (- 30) \cdot 0 = 15

(- 15)^{2} - 4 (- 30) \cdot 0

規則を適用

- (- a) = a

= (- 15)^{2} + 4 \cdot 30 \cdot 0

指数の規則を適用する:

n

が偶数であれば

(- a)^{n} = a^{n}

(- 15)^{2} = 1 5^{2}

= 1 5^{2} + 4 \cdot 30 \cdot 0

規則を適用

0 \cdot a = 0

= 1 5^{2} + 0

1 5^{2} + 0 = 1 5^{2}

= 1 5^{2}

累乗根の規則を適用する:

n a^{n} = a,

, 以下を想定

a \geq 0

= 15

u_{1, 2} = \frac{- ( - 15 ) \pm 15}{2 ( - 30 )}

解を分離する

u_{1} = \frac{- ( - 15 ) + 15}{2 ( - 30 )}, u_{2} = \frac{- ( - 15 ) - 15}{2 ( - 30 )}

u = \frac{- ( - 15 ) + 15}{2 ( - 30 )} : - \frac{1}{2}

\frac{- ( - 15 ) + 15}{2 ( - 30 )}

括弧を削除する:

(- a) = - a, - (- a) = a

= \frac{15 + 15}{- 2 \cdot 30}

数を足す:

15 + 15 = 30

= \frac{30}{- 2 \cdot 30}

数を乗じる:

2 \cdot 30 = 60

= \frac{30}{- 60}

分数の規則を適用する:

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

= - \frac{30}{60}

共通因数を約分する:

30

= - \frac{1}{2}

u = \frac{- ( - 15 ) - 15}{2 ( - 30 )} : 0

\frac{- ( - 15 ) - 15}{2 ( - 30 )}

括弧を削除する:

(- a) = - a, - (- a) = a

= \frac{15 - 15}{- 2 \cdot 30}

数を引く:

15 - 15 = 0

= \frac{0}{- 2 \cdot 30}

数を乗じる:

2 \cdot 30 = 60

= \frac{0}{- 60}

分数の規則を適用する:

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

= - \frac{0}{60}

規則を適用

\frac{0}{a} = 0, a \neq = 0

= - 0

= 0

二次equationの解:

u = - \frac{1}{2}, u = 0

代用を戻す

u = cos (x)

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

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

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) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

以下の一般解

cos (x) = 0

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

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

グラフ

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