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人気のある三角関数 >

3sin^2(x)+8cos^2(x)=6

解

3 sin^{2} (x) + 8 cos^{2} (x) = 6

解

x = 0.68471 \dots + 2 πn, x = π - 0.68471 \dots + 2 πn, x = - 0.68471 \dots + 2 πn, x = π + 0.68471 \dots + 2 πn

+1

度

x = 39.23152 \dots^{\circ} + 36 0^{\circ} n, x = 140.76847 \dots^{\circ} + 36 0^{\circ} n, x = - 39.23152 \dots^{\circ} + 36 0^{\circ} n, x = 219.23152 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

3 sin^{2} (x) + 8 cos^{2} (x) = 6

両辺から

6

を引く

3 sin^{2} (x) + 8 cos^{2} (x) - 6 = 0

三角関数の公式を使用して書き換える

- 6 + 3 sin^{2} (x) + 8 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)

= - 6 + 3 sin^{2} (x) + 8 (1 - sin^{2} (x))

簡素化

- 6 + 3 sin^{2} (x) + 8 (1 - sin^{2} (x)) : - 5 sin^{2} (x) + 2

- 6 + 3 sin^{2} (x) + 8 (1 - sin^{2} (x))

拡張

8 (1 - sin^{2} (x)) : 8 - 8 sin^{2} (x)

8 (1 - sin^{2} (x))

分配法則を適用する:

a (b - c) = ab - a c

a = 8, b = 1, c = sin^{2} (x)

= 8 \cdot 1 - 8 sin^{2} (x)

数を乗じる:

8 \cdot 1 = 8

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

= - 6 + 3 sin^{2} (x) + 8 - 8 sin^{2} (x)

簡素化

- 6 + 3 sin^{2} (x) + 8 - 8 sin^{2} (x) : - 5 sin^{2} (x) + 2

- 6 + 3 sin^{2} (x) + 8 - 8 sin^{2} (x)

条件のようなグループ

= 3 sin^{2} (x) - 8 sin^{2} (x) - 6 + 8

類似した元を足す:

3 sin^{2} (x) - 8 sin^{2} (x) = - 5 sin^{2} (x)

= - 5 sin^{2} (x) - 6 + 8

数を足す/引く:

- 6 + 8 = 2

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

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

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

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

置換で解く

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

仮定:

sin (x) = u

2 - 5 u^{2} = 0

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

2 - 5 u^{2} = 0

2

を右側に移動します

2 - 5 u^{2} = 0

両辺から

2

を引く

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

簡素化

- 5 u^{2} = - 2

- 5 u^{2} = - 2

以下で両辺を割る

- 5

- 5 u^{2} = - 2

以下で両辺を割る

- 5

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

簡素化

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

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

x^{2} = f (a)

の場合, 解は

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

u = \frac{2}{5}, u = - \frac{2}{5}

代用を戻す

u = sin (x)

sin (x) = \frac{2}{5}, sin (x) = - \frac{2}{5}

sin (x) = \frac{2}{5}, sin (x) = - \frac{2}{5}

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

sin (x) = \frac{2}{5}

三角関数の逆数プロパティを適用する

sin (x) = \frac{2}{5}

以下の一般解

sin (x) = \frac{2}{5}

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

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

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

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

sin (x) = - \frac{2}{5}

三角関数の逆数プロパティを適用する

sin (x) = - \frac{2}{5}

以下の一般解

sin (x) = - \frac{2}{5}

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

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

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

すべての解を組み合わせる

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

10進法形式で解を証明する

x = 0.68471 \dots + 2 πn, x = π - 0.68471 \dots + 2 πn, x = - 0.68471 \dots + 2 πn, x = π + 0.68471 \dots + 2 πn

グラフ

人気の例

tan (θ) = \frac{2}{8}

4tan^2(x)+7tan(x)-4=0,cot(2x)

4 tan^{2} (x) + 7 tan (x) - 4 = 0, cot (2 x)

11.75=5sin((pit)/6-(7pi)/6)+8

11.75 = 5 sin (\frac{π t}{6} - \frac{7 π}{6}) + 8

0=-2sin(x)-2cos(2x)

0 = - 2 sin (x) - 2 cos (2 x)

cot(a)= 7/(sqrt(51))

cot (a) = \frac{7}{51}