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人気のある三角関数 >

50sin(x)+15cos(x)=40,0<x<pi

解

50 sin (x) + 15 cos (x) = 40, 0 < x < π

解

x = 1.97713 \dots, x = 0.58154 \dots

+1

度

x = 113.28144 \dots^{\circ}, x = 33.32006 \dots^{\circ}

解答ステップ

50 sin (x) + 15 cos (x) = 40, 0 < x < π

両辺から

15 cos (x)

を引く

50 sin (x) = 40 - 15 cos (x)

両辺を2乗する

(50 sin (x))^{2} = (40 - 15 cos (x))^{2}

両辺から

(40 - 15 cos (x))^{2}

を引く

2500 sin^{2} (x) - 1600 + 1200 cos (x) - 225 cos^{2} (x) = 0

三角関数の公式を使用して書き換える

- 1600 + 1200 cos (x) - 225 cos^{2} (x) + 2500 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)

= - 1600 + 1200 cos (x) - 225 cos^{2} (x) + 2500 (1 - cos^{2} (x))

簡素化

- 1600 + 1200 cos (x) - 225 cos^{2} (x) + 2500 (1 - cos^{2} (x)) : 1200 cos (x) - 2725 cos^{2} (x) + 900

- 1600 + 1200 cos (x) - 225 cos^{2} (x) + 2500 (1 - cos^{2} (x))

拡張

2500 (1 - cos^{2} (x)) : 2500 - 2500 cos^{2} (x)

2500 (1 - cos^{2} (x))

分配法則を適用する:

a (b - c) = ab - a c

a = 2500, b = 1, c = cos^{2} (x)

= 2500 \cdot 1 - 2500 cos^{2} (x)

数を乗じる:

2500 \cdot 1 = 2500

= 2500 - 2500 cos^{2} (x)

= - 1600 + 1200 cos (x) - 225 cos^{2} (x) + 2500 - 2500 cos^{2} (x)

簡素化

- 1600 + 1200 cos (x) - 225 cos^{2} (x) + 2500 - 2500 cos^{2} (x) : 1200 cos (x) - 2725 cos^{2} (x) + 900

- 1600 + 1200 cos (x) - 225 cos^{2} (x) + 2500 - 2500 cos^{2} (x)

条件のようなグループ

= 1200 cos (x) - 225 cos^{2} (x) - 2500 cos^{2} (x) - 1600 + 2500

類似した元を足す:

- 225 cos^{2} (x) - 2500 cos^{2} (x) = - 2725 cos^{2} (x)

= 1200 cos (x) - 2725 cos^{2} (x) - 1600 + 2500

数を足す/引く:

- 1600 + 2500 = 900

= 1200 cos (x) - 2725 cos^{2} (x) + 900

= 1200 cos (x) - 2725 cos^{2} (x) + 900

= 1200 cos (x) - 2725 cos^{2} (x) + 900

900 + 1200 cos (x) - 2725 cos^{2} (x) = 0

置換で解く

900 + 1200 cos (x) - 2725 cos^{2} (x) = 0

仮定:

cos (x) = u

900 + 1200 u - 2725 u^{2} = 0

900 + 1200 u - 2725 u^{2} = 0 : u = - \frac{6 ( 5 5 - 4 )}{109}, u = \frac{6 ( 4 + 5 5 )}{109}

900 + 1200 u - 2725 u^{2} = 0

標準的な形式で書く

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

- 2725 u^{2} + 1200 u + 900 = 0

解くとthe二次式

- 2725 u^{2} + 1200 u + 900 = 0

二次Equationの公式:

次の場合:

a = - 2725, b = 1200, c = 900

u_{1, 2} = \frac{- 1200 \pm 120 0 ^{2} - 4 ( - 2725 ) \cdot 900}{2 ( - 2725 )}

u_{1, 2} = \frac{- 1200 \pm 120 0 ^{2} - 4 ( - 2725 ) \cdot 900}{2 ( - 2725 )}

120 0^{2} - 4 (- 2725) \cdot 900 = 15005

120 0^{2} - 4 (- 2725) \cdot 900

規則を適用

- (- a) = a

= 120 0^{2} + 4 \cdot 2725 \cdot 900

数を乗じる:

4 \cdot 2725 \cdot 900 = 9810000

= 120 0^{2} + 9810000

120 0^{2} = 1440000

= 1440000 + 9810000

数を足す:

1440000 + 9810000 = 11250000

= 11250000

以下の素因数分解:

11250000 : 2^{4} \cdot 3^{2} \cdot 5^{7}

11250000

= 5^{7} \cdot 2^{4} \cdot 3^{2}

指数の規則を適用する:

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

= 5^{6} \cdot 2^{4} \cdot 3^{2} \cdot 5

累乗根の規則を適用する:

n ab = n a n b

= 5 2^{4} 3^{2} 5^{6}

累乗根の規則を適用する:

n a^{m} = a^{\frac{m}{n}}

2^{4} = 2^{\frac{4}{2}} = 2^{2}

= 2^{2} 5 3^{2} 5^{6}

累乗根の規則を適用する:

n a^{n} = a

3^{2} = 3

= 2^{2} \cdot 35 5^{6}

累乗根の規則を適用する:

n a^{m} = a^{\frac{m}{n}}

5^{6} = 5^{\frac{6}{2}} = 5^{3}

= 5^{3} \cdot 2^{2} \cdot 35

改良

= 15005

u_{1, 2} = \frac{- 1200 \pm 1500 5}{2 ( - 2725 )}

解を分離する

u_{1} = \frac{- 1200 + 1500 5}{2 ( - 2725 )}, u_{2} = \frac{- 1200 - 1500 5}{2 ( - 2725 )}

u = \frac{- 1200 + 1500 5}{2 ( - 2725 )} : - \frac{6 ( 5 5 - 4 )}{109}

\frac{- 1200 + 1500 5}{2 ( - 2725 )}

括弧を削除する:

(- a) = - a

= \frac{- 1200 + 1500 5}{- 2 \cdot 2725}

数を乗じる:

2 \cdot 2725 = 5450

= \frac{- 1200 + 1500 5}{- 5450}

分数の規則を適用する:

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

= - \frac{- 1200 + 1500 5}{5450}

キャンセル

\frac{- 1200 + 1500 5}{5450} : \frac{6 ( 5 5 - 4 )}{109}

\frac{- 1200 + 1500 5}{5450}

因数

- 1200 + 15005 : 300 (- 4 + 55)

- 1200 + 15005

書き換え

= - 300 \cdot 4 + 300 \cdot 55

共通項をくくり出す

300

= 300 (- 4 + 55)

= \frac{300 ( - 4 + 5 5 )}{5450}

共通因数を約分する:

50

= \frac{6 ( 5 5 - 4 )}{109}

= - \frac{6 ( 5 5 - 4 )}{109}

u = \frac{- 1200 - 1500 5}{2 ( - 2725 )} : \frac{6 ( 4 + 5 5 )}{109}

\frac{- 1200 - 1500 5}{2 ( - 2725 )}

括弧を削除する:

(- a) = - a

= \frac{- 1200 - 1500 5}{- 2 \cdot 2725}

数を乗じる:

2 \cdot 2725 = 5450

= \frac{- 1200 - 1500 5}{- 5450}

分数の規則を適用する:

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

- 1200 - 15005 = - (1200 + 15005)

= \frac{1200 + 1500 5}{5450}

因数

1200 + 15005 : 300 (4 + 55)

1200 + 15005

書き換え

= 300 \cdot 4 + 300 \cdot 55

共通項をくくり出す

300

= 300 (4 + 55)

= \frac{300 ( 4 + 5 5 )}{5450}

共通因数を約分する:

50

= \frac{6 ( 4 + 5 5 )}{109}

二次equationの解:

u = - \frac{6 ( 5 5 - 4 )}{109}, u = \frac{6 ( 4 + 5 5 )}{109}

代用を戻す

u = cos (x)

cos (x) = - \frac{6 ( 5 5 - 4 )}{109}, cos (x) = \frac{6 ( 4 + 5 5 )}{109}

cos (x) = - \frac{6 ( 5 5 - 4 )}{109}, cos (x) = \frac{6 ( 4 + 5 5 )}{109}

cos (x) = - \frac{6 ( 5 5 - 4 )}{109}, 0 < x < π : x = arccos (- \frac{6 ( 5 5 - 4 )}{109})

cos (x) = - \frac{6 ( 5 5 - 4 )}{109}, 0 < x < π

三角関数の逆数プロパティを適用する

cos (x) = - \frac{6 ( 5 5 - 4 )}{109}

以下の一般解

cos (x) = - \frac{6 ( 5 5 - 4 )}{109}

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

x = arccos (- \frac{6 ( 5 5 - 4 )}{109}) + 2 πn, x = - arccos (- \frac{6 ( 5 5 - 4 )}{109}) + 2 πn

x = arccos (- \frac{6 ( 5 5 - 4 )}{109}) + 2 πn, x = - arccos (- \frac{6 ( 5 5 - 4 )}{109}) + 2 πn

範囲の解答

0 < x < π

x = arccos (- \frac{6 ( 5 5 - 4 )}{109})

cos (x) = \frac{6 ( 4 + 5 5 )}{109}, 0 < x < π : x = arccos (\frac{6 ( 4 + 5 5 )}{109})

cos (x) = \frac{6 ( 4 + 5 5 )}{109}, 0 < x < π

三角関数の逆数プロパティを適用する

cos (x) = \frac{6 ( 4 + 5 5 )}{109}

以下の一般解

cos (x) = \frac{6 ( 4 + 5 5 )}{109}

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

x = arccos (\frac{6 ( 4 + 5 5 )}{109}) + 2 πn, x = 2 π - arccos (\frac{6 ( 4 + 5 5 )}{109}) + 2 πn

x = arccos (\frac{6 ( 4 + 5 5 )}{109}) + 2 πn, x = 2 π - arccos (\frac{6 ( 4 + 5 5 )}{109}) + 2 πn

範囲の解答

0 < x < π

x = arccos (\frac{6 ( 4 + 5 5 )}{109})

すべての解を組み合わせる

x = arccos (- \frac{6 ( 5 5 - 4 )}{109}), x = arccos (\frac{6 ( 4 + 5 5 )}{109})

元のequationに当てはめて解を検算する

50 sin (x) + 15 cos (x) = 40

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{6 ( 5 5 - 4 )}{109}) :

真

arccos (- \frac{6 ( 5 5 - 4 )}{109})

挿入

n = 1

arccos (- \frac{6 ( 5 5 - 4 )}{109})

50 sin (x) + 15 cos (x) = 40

の挿入向け

x = arccos (- \frac{6 ( 5 5 - 4 )}{109})

50 sin (arccos (- \frac{6 ( 5 5 - 4 )}{109})) + 15 cos (arccos (- \frac{6 ( 5 5 - 4 )}{109})) = 40

改良

40 = 40

\Rightarrow 真

解答を確認する

arccos (\frac{6 ( 4 + 5 5 )}{109}) :

真

arccos (\frac{6 ( 4 + 5 5 )}{109})

挿入

n = 1

arccos (\frac{6 ( 4 + 5 5 )}{109})

50 sin (x) + 15 cos (x) = 40

の挿入向け

x = arccos (\frac{6 ( 4 + 5 5 )}{109})

50 sin (arccos (\frac{6 ( 4 + 5 5 )}{109})) + 15 cos (arccos (\frac{6 ( 4 + 5 5 )}{109})) = 40

改良

40 = 40

\Rightarrow 真

x = arccos (- \frac{6 ( 5 5 - 4 )}{109}), x = arccos (\frac{6 ( 4 + 5 5 )}{109})

10進法形式で解を証明する

x = 1.97713 \dots, x = 0.58154 \dots

グラフ

人気の例

10sin(x)cos(x)=6cos(x)

10 sin (x) cos (x) = 6 cos (x)

sin(x)+cos(x)=sec(x)

sin (x) + cos (x) = sec (x)

sin (x) = - 0.2761

sin (2 y) = 0

7sin^2(θ)-15sin(θ)+8=0

7 sin^{2} (θ) - 15 sin (θ) + 8 = 0