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人気のある三角関数 >

3sin(x)tan(x)=8

解

3 sin (x) tan (x) = 8

解

x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn

+1

度

x = 70.52877 \dots^{\circ} + 36 0^{\circ} n, x = 289.47122 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

3 sin (x) tan (x) = 8

両辺から

8

を引く

3 sin (x) tan (x) - 8 = 0

サイン, コサインで表わす

3 sin (x) \frac{sin ( x )}{cos ( x )} - 8 = 0

簡素化

3 sin (x) \frac{sin ( x )}{cos ( x )} - 8 : \frac{3 sin ^{2} ( x ) - 8 cos ( x )}{cos ( x )}

3 sin (x) \frac{sin ( x )}{cos ( x )} - 8

3 sin (x) \frac{sin ( x )}{cos ( x )} = \frac{3 sin ^{2} ( x )}{cos ( x )}

3 sin (x) \frac{sin ( x )}{cos ( x )}

分数を乗じる:

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

= \frac{sin ( x ) \cdot 3 sin ( x )}{cos ( x )}

sin (x) \cdot 3 sin (x) = 3 sin^{2} (x)

sin (x) \cdot 3 sin (x)

指数の規則を適用する:

a^{b} \cdot a^{c} = a^{b + c}

sin (x) sin (x) = sin^{1 + 1} (x)

= 3 sin^{1 + 1} (x)

数を足す:

1 + 1 = 2

= 3 sin^{2} (x)

= \frac{3 sin ^{2} ( x )}{cos ( x )}

= \frac{3 sin ^{2} ( x )}{cos ( x )} - 8

元を分数に変換する:

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

= \frac{3 sin ^{2} ( x )}{cos ( x )} - \frac{8 cos ( x )}{cos ( x )}

分母が等しいので, 分数を組み合わせる:

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

= \frac{3 sin ^{2} ( x ) - 8 cos ( x )}{cos ( x )}

\frac{3 sin ^{2} ( x ) - 8 cos ( x )}{cos ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

3 sin^{2} (x) - 8 cos (x) = 0

両辺に

8 cos (x)

を足す

3 sin^{2} (x) = 8 cos (x)

両辺を2乗する

(3 sin^{2} (x))^{2} = (8 cos (x))^{2}

両辺から

(8 cos (x))^{2}

を引く

9 sin^{4} (x) - 64 cos^{2} (x) = 0

因数

9 sin^{4} (x) - 64 cos^{2} (x) : (3 sin^{2} (x) + 8 cos (x)) (3 sin^{2} (x) - 8 cos (x))

9 sin^{4} (x) - 64 cos^{2} (x)

9 sin^{4} (x) - 64 cos^{2} (x)

を書き換え

(3 sin^{2} (x))^{2} - (8 cos (x))^{2}

9 sin^{4} (x) - 64 cos^{2} (x)

9

を書き換え

3^{2}

= 3^{2} sin^{4} (x) - 64 cos^{2} (x)

64

を書き換え

8^{2}

= 3^{2} sin^{4} (x) - 8^{2} cos^{2} (x)

指数の規則を適用する:

a^{b c} = (a^{b})^{c}

sin^{4} (x) = (sin^{2} (x))^{2}

= 3^{2} (sin^{2} (x))^{2} - 8^{2} cos^{2} (x)

指数の規則を適用する:

a^{m} b^{m} = (ab)^{m}

3^{2} (sin^{2} (x))^{2} = (3 sin^{2} (x))^{2}

= (3 sin^{2} (x))^{2} - 8^{2} cos^{2} (x)

指数の規則を適用する:

a^{m} b^{m} = (ab)^{m}

8^{2} cos^{2} (x) = (8 cos (x))^{2}

= (3 sin^{2} (x))^{2} - (8 cos (x))^{2}

= (3 sin^{2} (x))^{2} - (8 cos (x))^{2}

2乗の差の公式を適用する:

x^{2} - y^{2} = (x + y) (x - y)

(3 sin^{2} (x))^{2} - (8 cos (x))^{2} = (3 sin^{2} (x) + 8 cos (x)) (3 sin^{2} (x) - 8 cos (x))

= (3 sin^{2} (x) + 8 cos (x)) (3 sin^{2} (x) - 8 cos (x))

(3 sin^{2} (x) + 8 cos (x)) (3 sin^{2} (x) - 8 cos (x)) = 0

各部分を別個に解く

3 sin^{2} (x) + 8 cos (x) = 0 or 3 sin^{2} (x) - 8 cos (x) = 0

3 sin^{2} (x) + 8 cos (x) = 0 : x = arccos (- \frac{1}{3}) + 2 πn, x = - arccos (- \frac{1}{3}) + 2 πn

3 sin^{2} (x) + 8 cos (x) = 0

三角関数の公式を使用して書き換える

3 sin^{2} (x) + 8 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)

= 3 (1 - cos^{2} (x)) + 8 cos (x)

(1 - cos^{2} (x)) \cdot 3 + 8 cos (x) = 0

置換で解く

(1 - cos^{2} (x)) \cdot 3 + 8 cos (x) = 0

仮定:

cos (x) = u

(1 - u^{2}) \cdot 3 + 8 u = 0

(1 - u^{2}) \cdot 3 + 8 u = 0 : u = - \frac{1}{3}, u = 3

(1 - u^{2}) \cdot 3 + 8 u = 0

拡張

(1 - u^{2}) \cdot 3 + 8 u : 3 - 3 u^{2} + 8 u

(1 - u^{2}) \cdot 3 + 8 u

= 3 (1 - u^{2}) + 8 u

拡張

3 (1 - u^{2}) : 3 - 3 u^{2}

3 (1 - u^{2})

分配法則を適用する:

a (b - c) = ab - a c

a = 3, b = 1, c = u^{2}

= 3 \cdot 1 - 3 u^{2}

数を乗じる:

3 \cdot 1 = 3

= 3 - 3 u^{2}

= 3 - 3 u^{2} + 8 u

3 - 3 u^{2} + 8 u = 0

標準的な形式で書く

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

- 3 u^{2} + 8 u + 3 = 0

解くとthe二次式

- 3 u^{2} + 8 u + 3 = 0

二次Equationの公式:

次の場合:

a = - 3, b = 8, c = 3

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 ( - 3 ) \cdot 3}{2 ( - 3 )}

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 ( - 3 ) \cdot 3}{2 ( - 3 )}

8^{2} - 4 (- 3) \cdot 3 = 10

8^{2} - 4 (- 3) \cdot 3

規則を適用

- (- a) = a

= 8^{2} + 4 \cdot 3 \cdot 3

数を乗じる:

4 \cdot 3 \cdot 3 = 36

= 8^{2} + 36

8^{2} = 64

= 64 + 36

数を足す:

64 + 36 = 100

= 100

数を因数に分解する:

100 = 1 0^{2}

= 1 0^{2}

累乗根の規則を適用する:

n a^{n} = a

1 0^{2} = 10

= 10

u_{1, 2} = \frac{- 8 \pm 10}{2 ( - 3 )}

解を分離する

u_{1} = \frac{- 8 + 10}{2 ( - 3 )}, u_{2} = \frac{- 8 - 10}{2 ( - 3 )}

u = \frac{- 8 + 10}{2 ( - 3 )} : - \frac{1}{3}

\frac{- 8 + 10}{2 ( - 3 )}

括弧を削除する:

(- a) = - a

= \frac{- 8 + 10}{- 2 \cdot 3}

数を足す/引く:

- 8 + 10 = 2

= \frac{2}{- 2 \cdot 3}

数を乗じる:

2 \cdot 3 = 6

= \frac{2}{- 6}

分数の規則を適用する:

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

= - \frac{2}{6}

共通因数を約分する:

2

= - \frac{1}{3}

u = \frac{- 8 - 10}{2 ( - 3 )} : 3

\frac{- 8 - 10}{2 ( - 3 )}

括弧を削除する:

(- a) = - a

= \frac{- 8 - 10}{- 2 \cdot 3}

数を引く:

- 8 - 10 = - 18

= \frac{- 18}{- 2 \cdot 3}

数を乗じる:

2 \cdot 3 = 6

= \frac{- 18}{- 6}

分数の規則を適用する:

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

= \frac{18}{6}

数を割る:

\frac{18}{6} = 3

= 3

二次equationの解:

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

代用を戻す

u = cos (x)

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

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

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

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

三角関数の逆数プロパティを適用する

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

以下の一般解

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

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

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

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

cos (x) = 3 :

解なし

cos (x) = 3

- 1 \leq cos (x) \leq 1

解なし

すべての解を組み合わせる

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

3 sin^{2} (x) - 8 cos (x) = 0 : x = arccos (\frac{1}{3}) + 2 πn, x = 2 π - arccos (\frac{1}{3}) + 2 πn

3 sin^{2} (x) - 8 cos (x) = 0

三角関数の公式を使用して書き換える

3 sin^{2} (x) - 8 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)

= 3 (1 - cos^{2} (x)) - 8 cos (x)

(1 - cos^{2} (x)) \cdot 3 - 8 cos (x) = 0

置換で解く

(1 - cos^{2} (x)) \cdot 3 - 8 cos (x) = 0

仮定:

cos (x) = u

(1 - u^{2}) \cdot 3 - 8 u = 0

(1 - u^{2}) \cdot 3 - 8 u = 0 : u = - 3, u = \frac{1}{3}

(1 - u^{2}) \cdot 3 - 8 u = 0

拡張

(1 - u^{2}) \cdot 3 - 8 u : 3 - 3 u^{2} - 8 u

(1 - u^{2}) \cdot 3 - 8 u

= 3 (1 - u^{2}) - 8 u

拡張

3 (1 - u^{2}) : 3 - 3 u^{2}

3 (1 - u^{2})

分配法則を適用する:

a (b - c) = ab - a c

a = 3, b = 1, c = u^{2}

= 3 \cdot 1 - 3 u^{2}

数を乗じる:

3 \cdot 1 = 3

= 3 - 3 u^{2}

= 3 - 3 u^{2} - 8 u

3 - 3 u^{2} - 8 u = 0

標準的な形式で書く

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

- 3 u^{2} - 8 u + 3 = 0

解くとthe二次式

- 3 u^{2} - 8 u + 3 = 0

二次Equationの公式:

次の場合:

a = - 3, b = - 8, c = 3

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 ( - 3 ) \cdot 3}{2 ( - 3 )}

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 ( - 3 ) \cdot 3}{2 ( - 3 )}

(- 8)^{2} - 4 (- 3) \cdot 3 = 10

(- 8)^{2} - 4 (- 3) \cdot 3

規則を適用

- (- a) = a

= (- 8)^{2} + 4 \cdot 3 \cdot 3

指数の規則を適用する:

n

が偶数であれば

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

(- 8)^{2} = 8^{2}

= 8^{2} + 4 \cdot 3 \cdot 3

数を乗じる:

4 \cdot 3 \cdot 3 = 36

= 8^{2} + 36

8^{2} = 64

= 64 + 36

数を足す:

64 + 36 = 100

= 100

数を因数に分解する:

100 = 1 0^{2}

= 1 0^{2}

累乗根の規則を適用する:

n a^{n} = a

1 0^{2} = 10

= 10

u_{1, 2} = \frac{- ( - 8 ) \pm 10}{2 ( - 3 )}

解を分離する

u_{1} = \frac{- ( - 8 ) + 10}{2 ( - 3 )}, u_{2} = \frac{- ( - 8 ) - 10}{2 ( - 3 )}

u = \frac{- ( - 8 ) + 10}{2 ( - 3 )} : - 3

\frac{- ( - 8 ) + 10}{2 ( - 3 )}

括弧を削除する:

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

= \frac{8 + 10}{- 2 \cdot 3}

数を足す:

8 + 10 = 18

= \frac{18}{- 2 \cdot 3}

数を乗じる:

2 \cdot 3 = 6

= \frac{18}{- 6}

分数の規則を適用する:

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

= - \frac{18}{6}

数を割る:

\frac{18}{6} = 3

= - 3

u = \frac{- ( - 8 ) - 10}{2 ( - 3 )} : \frac{1}{3}

\frac{- ( - 8 ) - 10}{2 ( - 3 )}

括弧を削除する:

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

= \frac{8 - 10}{- 2 \cdot 3}

数を引く:

8 - 10 = - 2

= \frac{- 2}{- 2 \cdot 3}

数を乗じる:

2 \cdot 3 = 6

= \frac{- 2}{- 6}

分数の規則を適用する:

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

= \frac{2}{6}

共通因数を約分する:

2

= \frac{1}{3}

二次equationの解:

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

代用を戻す

u = cos (x)

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

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

cos (x) = - 3 :

解なし

cos (x) = - 3

- 1 \leq cos (x) \leq 1

解なし

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

cos (x) = \frac{1}{3}

三角関数の逆数プロパティを適用する

cos (x) = \frac{1}{3}

以下の一般解

cos (x) = \frac{1}{3}

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

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

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

すべての解を組み合わせる

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

すべての解を組み合わせる

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

元のequationに当てはめて解を検算する

3 sin (x) tan (x) = 8

に当てはめて解を確認する

equationに一致しないものを削除する。

解答を確認する

arccos (- \frac{1}{3}) + 2 πn :

偽

arccos (- \frac{1}{3}) + 2 πn

挿入

n = 1

arccos (- \frac{1}{3}) + 2 π 1

3 sin (x) tan (x) = 8

の挿入向け

x = arccos (- \frac{1}{3}) + 2 π 1

3 sin (arccos (- \frac{1}{3}) + 2 π 1) tan (arccos (- \frac{1}{3}) + 2 π 1) = 8

改良

- 8 = 8

\Rightarrow 偽

解答を確認する

- arccos (- \frac{1}{3}) + 2 πn :

偽

- arccos (- \frac{1}{3}) + 2 πn

挿入

n = 1

- arccos (- \frac{1}{3}) + 2 π 1

3 sin (x) tan (x) = 8

の挿入向け

x = - arccos (- \frac{1}{3}) + 2 π 1

3 sin (- arccos (- \frac{1}{3}) + 2 π 1) tan (- arccos (- \frac{1}{3}) + 2 π 1) = 8

改良

- 8 = 8

\Rightarrow 偽

解答を確認する

arccos (\frac{1}{3}) + 2 πn :

真

arccos (\frac{1}{3}) + 2 πn

挿入

n = 1

arccos (\frac{1}{3}) + 2 π 1

3 sin (x) tan (x) = 8

の挿入向け

x = arccos (\frac{1}{3}) + 2 π 1

3 sin (arccos (\frac{1}{3}) + 2 π 1) tan (arccos (\frac{1}{3}) + 2 π 1) = 8

改良

8 = 8

\Rightarrow 真

解答を確認する

2 π - arccos (\frac{1}{3}) + 2 πn :

真

2 π - arccos (\frac{1}{3}) + 2 πn

挿入

n = 1

2 π - arccos (\frac{1}{3}) + 2 π 1

3 sin (x) tan (x) = 8

の挿入向け

x = 2 π - arccos (\frac{1}{3}) + 2 π 1

3 sin (2 π - arccos (\frac{1}{3}) + 2 π 1) tan (2 π - arccos (\frac{1}{3}) + 2 π 1) = 8

改良

8 = 8

\Rightarrow 真

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

10進法形式で解を証明する

x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn

グラフ

人気の例

tan^{2} (x) - 4 = 0

2sin^2(θ)+sin(θ)-1=0,0<= θ<2pi

2 sin^{2} (θ) + sin (θ) - 1 = 0, 0 \leq θ < 2 π

2cos^2(x)-3cos(x)=2

2 cos^{2} (x) - 3 cos (x) = 2

cot(θ)= 1/(sqrt(3))

cot (θ) = \frac{1}{3}

-3sin^2(θ)+4sin(θ)-7=-6

- 3 sin^{2} (θ) + 4 sin (θ) - 7 = - 6