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人気のある三角関数 >

(1+sin^2(x))=(4cos(x))^2

解

(1 + sin^{2} (x)) = (4 cos (x))^{2}

解

x = 1.22069 \dots + 2 πn, x = π - 1.22069 \dots + 2 πn, x = - 1.22069 \dots + 2 πn, x = π + 1.22069 \dots + 2 πn

+1

度

x = 69.94041 \dots^{\circ} + 36 0^{\circ} n, x = 110.05958 \dots^{\circ} + 36 0^{\circ} n, x = - 69.94041 \dots^{\circ} + 36 0^{\circ} n, x = 249.94041 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

(1 + sin^{2} (x)) = (4 cos (x))^{2}

両辺から

(4 cos (x))^{2}

を引く

1 + sin^{2} (x) - 16 cos^{2} (x) = 0

三角関数の公式を使用して書き換える

1 + sin^{2} (x) - 16 cos^{2} (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)

= 1 + sin^{2} (x) - 16 (1 - sin^{2} (x))

簡素化

1 + sin^{2} (x) - 16 (1 - sin^{2} (x)) : 17 sin^{2} (x) - 15

1 + sin^{2} (x) - 16 (1 - sin^{2} (x))

拡張

- 16 (1 - sin^{2} (x)) : - 16 + 16 sin^{2} (x)

- 16 (1 - sin^{2} (x))

分配法則を適用する:

a (b - c) = ab - a c

a = - 16, b = 1, c = sin^{2} (x)

= - 16 \cdot 1 - (- 16) sin^{2} (x)

マイナス・プラスの規則を適用する

- (- a) = a

= - 16 \cdot 1 + 16 sin^{2} (x)

数を乗じる:

16 \cdot 1 = 16

= - 16 + 16 sin^{2} (x)

= 1 + sin^{2} (x) - 16 + 16 sin^{2} (x)

簡素化

1 + sin^{2} (x) - 16 + 16 sin^{2} (x) : 17 sin^{2} (x) - 15

1 + sin^{2} (x) - 16 + 16 sin^{2} (x)

条件のようなグループ

= sin^{2} (x) + 16 sin^{2} (x) + 1 - 16

類似した元を足す:

sin^{2} (x) + 16 sin^{2} (x) = 17 sin^{2} (x)

= 17 sin^{2} (x) + 1 - 16

数を足す/引く:

1 - 16 = - 15

= 17 sin^{2} (x) - 15

= 17 sin^{2} (x) - 15

= 17 sin^{2} (x) - 15

- 15 + 17 sin^{2} (x) = 0

置換で解く

- 15 + 17 sin^{2} (x) = 0

仮定:

sin (x) = u

- 15 + 17 u^{2} = 0

- 15 + 17 u^{2} = 0 : u = \frac{15}{17}, u = - \frac{15}{17}

- 15 + 17 u^{2} = 0

15

を右側に移動します

- 15 + 17 u^{2} = 0

両辺に

15

を足す

- 15 + 17 u^{2} + 15 = 0 + 15

簡素化

17 u^{2} = 15

17 u^{2} = 15

以下で両辺を割る

17

17 u^{2} = 15

以下で両辺を割る

17

\frac{17 u ^{2}}{17} = \frac{15}{17}

簡素化

u^{2} = \frac{15}{17}

u^{2} = \frac{15}{17}

x^{2} = f (a)

の場合, 解は

x = f (a), - f (a)

u = \frac{15}{17}, u = - \frac{15}{17}

代用を戻す

u = sin (x)

sin (x) = \frac{15}{17}, sin (x) = - \frac{15}{17}

sin (x) = \frac{15}{17}, sin (x) = - \frac{15}{17}

sin (x) = \frac{15}{17} : x = arcsin (\frac{15}{17}) + 2 πn, x = π - arcsin (\frac{15}{17}) + 2 πn

sin (x) = \frac{15}{17}

三角関数の逆数プロパティを適用する

sin (x) = \frac{15}{17}

以下の一般解

sin (x) = \frac{15}{17}

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

x = arcsin (\frac{15}{17}) + 2 πn, x = π - arcsin (\frac{15}{17}) + 2 πn

x = arcsin (\frac{15}{17}) + 2 πn, x = π - arcsin (\frac{15}{17}) + 2 πn

sin (x) = - \frac{15}{17} : x = arcsin (- \frac{15}{17}) + 2 πn, x = π + arcsin (\frac{15}{17}) + 2 πn

sin (x) = - \frac{15}{17}

三角関数の逆数プロパティを適用する

sin (x) = - \frac{15}{17}

以下の一般解

sin (x) = - \frac{15}{17}

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

x = arcsin (- \frac{15}{17}) + 2 πn, x = π + arcsin (\frac{15}{17}) + 2 πn

x = arcsin (- \frac{15}{17}) + 2 πn, x = π + arcsin (\frac{15}{17}) + 2 πn

すべての解を組み合わせる

x = arcsin (\frac{15}{17}) + 2 πn, x = π - arcsin (\frac{15}{17}) + 2 πn, x = arcsin (- \frac{15}{17}) + 2 πn, x = π + arcsin (\frac{15}{17}) + 2 πn

10進法形式で解を証明する

x = 1.22069 \dots + 2 πn, x = π - 1.22069 \dots + 2 πn, x = - 1.22069 \dots + 2 πn, x = π + 1.22069 \dots + 2 πn

グラフ

人気の例

sin^2(a)=(1-cos(a))/2

cos(x)=(-11)/(12)

6sin^2(x)+5cos(x)-2=0

(sin^{22}(a))/(sin^2(a))=4-4sin^2(a)