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人気のある三角関数 >

12sin^2(t)+cos(t)-4=2

解

12 sin^{2} (t) + cos (t) - 4 = 2

解

t = 2.30052 \dots + 2 πn, t = - 2.30052 \dots + 2 πn, t = 0.72273 \dots + 2 πn, t = 2 π - 0.72273 \dots + 2 πn

+1

度

t = 131.81031 \dots^{\circ} + 36 0^{\circ} n, t = - 131.81031 \dots^{\circ} + 36 0^{\circ} n, t = 41.40962 \dots^{\circ} + 36 0^{\circ} n, t = 318.59037 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

12 sin^{2} (t) + cos (t) - 4 = 2

両辺から

2

を引く

12 sin^{2} (t) + cos (t) - 6 = 0

三角関数の公式を使用して書き換える

- 6 + cos (t) + 12 sin^{2} (t)

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)

= - 6 + cos (t) + 12 (1 - cos^{2} (t))

簡素化

- 6 + cos (t) + 12 (1 - cos^{2} (t)) : cos (t) - 12 cos^{2} (t) + 6

- 6 + cos (t) + 12 (1 - cos^{2} (t))

拡張

12 (1 - cos^{2} (t)) : 12 - 12 cos^{2} (t)

12 (1 - cos^{2} (t))

分配法則を適用する:

a (b - c) = ab - a c

a = 12, b = 1, c = cos^{2} (t)

= 12 \cdot 1 - 12 cos^{2} (t)

数を乗じる:

12 \cdot 1 = 12

= 12 - 12 cos^{2} (t)

= - 6 + cos (t) + 12 - 12 cos^{2} (t)

簡素化

- 6 + cos (t) + 12 - 12 cos^{2} (t) : cos (t) - 12 cos^{2} (t) + 6

- 6 + cos (t) + 12 - 12 cos^{2} (t)

条件のようなグループ

= cos (t) - 12 cos^{2} (t) - 6 + 12

数を足す/引く:

- 6 + 12 = 6

= cos (t) - 12 cos^{2} (t) + 6

= cos (t) - 12 cos^{2} (t) + 6

= cos (t) - 12 cos^{2} (t) + 6

6 + cos (t) - 12 cos^{2} (t) = 0

置換で解く

6 + cos (t) - 12 cos^{2} (t) = 0

仮定:

cos (t) = u

6 + u - 12 u^{2} = 0

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

6 + u - 12 u^{2} = 0

標準的な形式で書く

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

- 12 u^{2} + u + 6 = 0

解くとthe二次式

- 12 u^{2} + u + 6 = 0

二次Equationの公式:

次の場合:

a = - 12, b = 1, c = 6

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 ( - 12 ) \cdot 6}{2 ( - 12 )}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 ( - 12 ) \cdot 6}{2 ( - 12 )}

1^{2} - 4 (- 12) \cdot 6 = 17

1^{2} - 4 (- 12) \cdot 6

規則を適用

1^{a} = 1

1^{2} = 1

= 1 - 4 (- 12) \cdot 6

規則を適用

- (- a) = a

= 1 + 4 \cdot 12 \cdot 6

数を乗じる:

4 \cdot 12 \cdot 6 = 288

= 1 + 288

数を足す:

1 + 288 = 289

= 289

数を因数に分解する:

289 = 1 7^{2}

= 1 7^{2}

累乗根の規則を適用する:

n a^{n} = a

1 7^{2} = 17

= 17

u_{1, 2} = \frac{- 1 \pm 17}{2 ( - 12 )}

解を分離する

u_{1} = \frac{- 1 + 17}{2 ( - 12 )}, u_{2} = \frac{- 1 - 17}{2 ( - 12 )}

u = \frac{- 1 + 17}{2 ( - 12 )} : - \frac{2}{3}

\frac{- 1 + 17}{2 ( - 12 )}

括弧を削除する:

(- a) = - a

= \frac{- 1 + 17}{- 2 \cdot 12}

数を足す/引く:

- 1 + 17 = 16

= \frac{16}{- 2 \cdot 12}

数を乗じる:

2 \cdot 12 = 24

= \frac{16}{- 24}

分数の規則を適用する:

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

= - \frac{16}{24}

共通因数を約分する:

8

= - \frac{2}{3}

u = \frac{- 1 - 17}{2 ( - 12 )} : \frac{3}{4}

\frac{- 1 - 17}{2 ( - 12 )}

括弧を削除する:

(- a) = - a

= \frac{- 1 - 17}{- 2 \cdot 12}

数を引く:

- 1 - 17 = - 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{2}{3}, u = \frac{3}{4}

代用を戻す

u = cos (t)

cos (t) = - \frac{2}{3}, cos (t) = \frac{3}{4}

cos (t) = - \frac{2}{3}, cos (t) = \frac{3}{4}

cos (t) = - \frac{2}{3} : t = arccos (- \frac{2}{3}) + 2 πn, t = - arccos (- \frac{2}{3}) + 2 πn

cos (t) = - \frac{2}{3}

三角関数の逆数プロパティを適用する

cos (t) = - \frac{2}{3}

以下の一般解

cos (t) = - \frac{2}{3}

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

t = arccos (- \frac{2}{3}) + 2 πn, t = - arccos (- \frac{2}{3}) + 2 πn

t = arccos (- \frac{2}{3}) + 2 πn, t = - arccos (- \frac{2}{3}) + 2 πn

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

cos (t) = \frac{3}{4}

三角関数の逆数プロパティを適用する

cos (t) = \frac{3}{4}

以下の一般解

cos (t) = \frac{3}{4}

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

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

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

すべての解を組み合わせる

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

10進法形式で解を証明する

t = 2.30052 \dots + 2 πn, t = - 2.30052 \dots + 2 πn, t = 0.72273 \dots + 2 πn, t = 2 π - 0.72273 \dots + 2 πn

グラフ

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