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人気のある三角関数 >

sin^2(x)+1= 7/2 cos^2(x)

解

sin^{2} (x) + 1 = \frac{7}{2} cos^{2} (x)

解

x = 0.84106 \dots + 2 πn, x = π - 0.84106 \dots + 2 πn, x = - 0.84106 \dots + 2 πn, x = π + 0.84106 \dots + 2 πn

+1

度

x = 48.18968 \dots^{\circ} + 36 0^{\circ} n, x = 131.81031 \dots^{\circ} + 36 0^{\circ} n, x = - 48.18968 \dots^{\circ} + 36 0^{\circ} n, x = 228.18968 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

sin^{2} (x) + 1 = \frac{7}{2} cos^{2} (x)

両辺から

\frac{7}{2} cos^{2} (x)

を引く

sin^{2} (x) + 1 - \frac{7}{2} cos^{2} (x) = 0

簡素化

sin^{2} (x) + 1 - \frac{7}{2} cos^{2} (x) : \frac{2 sin ^{2} ( x ) + 2 - 7 cos ^{2} ( x )}{2}

sin^{2} (x) + 1 - \frac{7}{2} cos^{2} (x)

乗じる

\frac{7}{2} cos^{2} (x) : \frac{7 cos ^{2} ( x )}{2}

\frac{7}{2} cos^{2} (x)

分数を乗じる:

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

= \frac{7 cos ^{2} ( x )}{2}

= sin^{2} (x) + 1 - \frac{7 cos ^{2} ( x )}{2}

元を分数に変換する:

sin^{2} (x) = \frac{sin ^{2} ( x ) 2}{2}, 1 = \frac{1 \cdot 2}{2}

= \frac{sin ^{2} ( x ) \cdot 2}{2} + \frac{1 \cdot 2}{2} - \frac{7 cos ^{2} ( x )}{2}

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{sin ^{2} ( x ) \cdot 2 + 1 \cdot 2 - 7 cos ^{2} ( x )}{2}

数を乗じる:

1 \cdot 2 = 2

= \frac{2 sin ^{2} ( x ) + 2 - 7 cos ^{2} ( x )}{2}

\frac{2 sin ^{2} ( x ) + 2 - 7 cos ^{2} ( x )}{2} = 0

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

2 sin^{2} (x) + 2 - 7 cos^{2} (x) = 0

三角関数の公式を使用して書き換える

2 + 2 sin^{2} (x) - 7 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)

= 2 + 2 sin^{2} (x) - 7 (1 - sin^{2} (x))

簡素化

2 + 2 sin^{2} (x) - 7 (1 - sin^{2} (x)) : 9 sin^{2} (x) - 5

2 + 2 sin^{2} (x) - 7 (1 - sin^{2} (x))

拡張

- 7 (1 - sin^{2} (x)) : - 7 + 7 sin^{2} (x)

- 7 (1 - sin^{2} (x))

分配法則を適用する:

a (b - c) = ab - a c

a = - 7, b = 1, c = sin^{2} (x)

= - 7 \cdot 1 - (- 7) sin^{2} (x)

マイナス・プラスの規則を適用する

- (- a) = a

= - 7 \cdot 1 + 7 sin^{2} (x)

数を乗じる:

7 \cdot 1 = 7

= - 7 + 7 sin^{2} (x)

= 2 + 2 sin^{2} (x) - 7 + 7 sin^{2} (x)

簡素化

2 + 2 sin^{2} (x) - 7 + 7 sin^{2} (x) : 9 sin^{2} (x) - 5

2 + 2 sin^{2} (x) - 7 + 7 sin^{2} (x)

条件のようなグループ

= 2 sin^{2} (x) + 7 sin^{2} (x) + 2 - 7

類似した元を足す:

2 sin^{2} (x) + 7 sin^{2} (x) = 9 sin^{2} (x)

= 9 sin^{2} (x) + 2 - 7

数を足す/引く:

2 - 7 = - 5

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

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

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

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

置換で解く

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

仮定:

sin (x) = u

- 5 + 9 u^{2} = 0

- 5 + 9 u^{2} = 0 : u = \frac{5}{3}, u = - \frac{5}{3}

- 5 + 9 u^{2} = 0

5

を右側に移動します

- 5 + 9 u^{2} = 0

両辺に

5

を足す

- 5 + 9 u^{2} + 5 = 0 + 5

簡素化

9 u^{2} = 5

9 u^{2} = 5

以下で両辺を割る

9

9 u^{2} = 5

以下で両辺を割る

9

\frac{9 u ^{2}}{9} = \frac{5}{9}

簡素化

u^{2} = \frac{5}{9}

u^{2} = \frac{5}{9}

x^{2} = f (a)

の場合, 解は

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

u = \frac{5}{9}, u = - \frac{5}{9}

\frac{5}{9} = \frac{5}{3}

\frac{5}{9}

累乗根の規則を適用する:

n \frac{a}{b} = \frac{n a}{n b},

, 以下を想定

a \geq 0, b \geq 0

= \frac{5}{9}

9 = 3

9

数を因数に分解する:

9 = 3^{2}

= 3^{2}

累乗根の規則を適用する:

n a^{n} = a

3^{2} = 3

= 3

= \frac{5}{3}

- \frac{5}{9} = - \frac{5}{3}

- \frac{5}{9}

簡素化

\frac{5}{9} : \frac{5}{3}

\frac{5}{9}

累乗根の規則を適用する:

n \frac{a}{b} = \frac{n a}{n b},

, 以下を想定

a \geq 0, b \geq 0

= \frac{5}{9}

9 = 3

9

数を因数に分解する:

9 = 3^{2}

= 3^{2}

累乗根の規則を適用する:

n a^{n} = a

3^{2} = 3

= 3

= \frac{5}{3}

= - \frac{5}{3}

u = \frac{5}{3}, u = - \frac{5}{3}

代用を戻す

u = sin (x)

sin (x) = \frac{5}{3}, sin (x) = - \frac{5}{3}

sin (x) = \frac{5}{3}, sin (x) = - \frac{5}{3}

sin (x) = \frac{5}{3} : x = arcsin (\frac{5}{3}) + 2 πn, x = π - arcsin (\frac{5}{3}) + 2 πn

sin (x) = \frac{5}{3}

三角関数の逆数プロパティを適用する

sin (x) = \frac{5}{3}

以下の一般解

sin (x) = \frac{5}{3}

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

x = arcsin (\frac{5}{3}) + 2 πn, x = π - arcsin (\frac{5}{3}) + 2 πn

x = arcsin (\frac{5}{3}) + 2 πn, x = π - arcsin (\frac{5}{3}) + 2 πn

sin (x) = - \frac{5}{3} : x = arcsin (- \frac{5}{3}) + 2 πn, x = π + arcsin (\frac{5}{3}) + 2 πn

sin (x) = - \frac{5}{3}

三角関数の逆数プロパティを適用する

sin (x) = - \frac{5}{3}

以下の一般解

sin (x) = - \frac{5}{3}

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

x = arcsin (- \frac{5}{3}) + 2 πn, x = π + arcsin (\frac{5}{3}) + 2 πn

x = arcsin (- \frac{5}{3}) + 2 πn, x = π + arcsin (\frac{5}{3}) + 2 πn

すべての解を組み合わせる

x = arcsin (\frac{5}{3}) + 2 πn, x = π - arcsin (\frac{5}{3}) + 2 πn, x = arcsin (- \frac{5}{3}) + 2 πn, x = π + arcsin (\frac{5}{3}) + 2 πn

10進法形式で解を証明する

x = 0.84106 \dots + 2 πn, x = π - 0.84106 \dots + 2 πn, x = - 0.84106 \dots + 2 πn, x = π + 0.84106 \dots + 2 πn

グラフ

人気の例

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tan(θ)=(sqrt(7))/3

tan (θ) = \frac{7}{3}

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