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人気のある三角関数 >

12cos^2(x)+5sin(x)-9=0

解

12 cos^{2} (x) + 5 sin (x) - 9 = 0

解

x = - 0.33983 \dots + 2 πn, x = π + 0.33983 \dots + 2 πn, x = 0.84806 \dots + 2 πn, x = π - 0.84806 \dots + 2 πn

+1

度

x = - 19.47122 \dots^{\circ} + 36 0^{\circ} n, x = 199.47122 \dots^{\circ} + 36 0^{\circ} n, x = 48.59037 \dots^{\circ} + 36 0^{\circ} n, x = 131.40962 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

12 cos^{2} (x) + 5 sin (x) - 9 = 0

三角関数の公式を使用して書き換える

- 9 + 12 cos^{2} (x) + 5 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 + 12 (1 - sin^{2} (x)) + 5 sin (x)

簡素化

- 9 + 12 (1 - sin^{2} (x)) + 5 sin (x) : 5 sin (x) - 12 sin^{2} (x) + 3

- 9 + 12 (1 - sin^{2} (x)) + 5 sin (x)

拡張

12 (1 - sin^{2} (x)) : 12 - 12 sin^{2} (x)

12 (1 - sin^{2} (x))

分配法則を適用する:

a (b - c) = ab - a c

a = 12, b = 1, c = sin^{2} (x)

= 12 \cdot 1 - 12 sin^{2} (x)

数を乗じる:

12 \cdot 1 = 12

= 12 - 12 sin^{2} (x)

= - 9 + 12 - 12 sin^{2} (x) + 5 sin (x)

数を足す/引く:

- 9 + 12 = 3

= 5 sin (x) - 12 sin^{2} (x) + 3

= 5 sin (x) - 12 sin^{2} (x) + 3

3 - 12 sin^{2} (x) + 5 sin (x) = 0

置換で解く

3 - 12 sin^{2} (x) + 5 sin (x) = 0

仮定:

sin (x) = u

3 - 12 u^{2} + 5 u = 0

3 - 12 u^{2} + 5 u = 0 : u = - \frac{1}{3}, u = \frac{3}{4}

3 - 12 u^{2} + 5 u = 0

標準的な形式で書く

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

- 12 u^{2} + 5 u + 3 = 0

解くとthe二次式

- 12 u^{2} + 5 u + 3 = 0

二次Equationの公式:

次の場合:

a = - 12, b = 5, c = 3

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 ( - 12 ) \cdot 3}{2 ( - 12 )}

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 ( - 12 ) \cdot 3}{2 ( - 12 )}

5^{2} - 4 (- 12) \cdot 3 = 13

5^{2} - 4 (- 12) \cdot 3

規則を適用

- (- a) = a

= 5^{2} + 4 \cdot 12 \cdot 3

数を乗じる:

4 \cdot 12 \cdot 3 = 144

= 5^{2} + 144

5^{2} = 25

= 25 + 144

数を足す:

25 + 144 = 169

= 169

数を因数に分解する:

169 = 1 3^{2}

= 1 3^{2}

累乗根の規則を適用する:

n a^{n} = a

1 3^{2} = 13

= 13

u_{1, 2} = \frac{- 5 \pm 13}{2 ( - 12 )}

解を分離する

u_{1} = \frac{- 5 + 13}{2 ( - 12 )}, u_{2} = \frac{- 5 - 13}{2 ( - 12 )}

u = \frac{- 5 + 13}{2 ( - 12 )} : - \frac{1}{3}

\frac{- 5 + 13}{2 ( - 12 )}

括弧を削除する:

(- a) = - a

= \frac{- 5 + 13}{- 2 \cdot 12}

数を足す/引く:

- 5 + 13 = 8

= \frac{8}{- 2 \cdot 12}

数を乗じる:

2 \cdot 12 = 24

= \frac{8}{- 24}

分数の規則を適用する:

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

= - \frac{8}{24}

共通因数を約分する:

8

= - \frac{1}{3}

u = \frac{- 5 - 13}{2 ( - 12 )} : \frac{3}{4}

\frac{- 5 - 13}{2 ( - 12 )}

括弧を削除する:

(- a) = - a

= \frac{- 5 - 13}{- 2 \cdot 12}

数を引く:

- 5 - 13 = - 18

= \frac{- 18}{- 2 \cdot 12}

数を乗じる:

2 \cdot 12 = 24

= \frac{- 18}{- 24}

分数の規則を適用する:

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

= \frac{18}{24}

共通因数を約分する:

6

= \frac{3}{4}

二次equationの解:

u = - \frac{1}{3}, u = \frac{3}{4}

代用を戻す

u = sin (x)

sin (x) = - \frac{1}{3}, sin (x) = \frac{3}{4}

sin (x) = - \frac{1}{3}, sin (x) = \frac{3}{4}

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

sin (x) = - \frac{1}{3}

三角関数の逆数プロパティを適用する

sin (x) = - \frac{1}{3}

以下の一般解

sin (x) = - \frac{1}{3}

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

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

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

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

sin (x) = \frac{3}{4}

三角関数の逆数プロパティを適用する

sin (x) = \frac{3}{4}

以下の一般解

sin (x) = \frac{3}{4}

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

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

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

すべての解を組み合わせる

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

10進法形式で解を証明する

x = - 0.33983 \dots + 2 πn, x = π + 0.33983 \dots + 2 πn, x = 0.84806 \dots + 2 πn, x = π - 0.84806 \dots + 2 πn

グラフ

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