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人気のある三角関数 >

3sin^2(x)-cos^2(x)=1

解

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

解

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

+1

度

x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

解答ステップ

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

両辺から

1

を引く

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

三角関数の公式を使用して書き換える

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

簡素化

- 1 - (1 - sin^{2} (x)) + 3 sin^{2} (x) : 4 sin^{2} (x) - 2

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

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

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

括弧を分配する

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

マイナス・プラスの規則を適用する

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

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

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

簡素化

- 1 - 1 + sin^{2} (x) + 3 sin^{2} (x) : 4 sin^{2} (x) - 2

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

類似した元を足す:

sin^{2} (x) + 3 sin^{2} (x) = 4 sin^{2} (x)

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

数を引く:

- 1 - 1 = - 2

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

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

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

- 2 + 4 sin^{2} (x) = 0

置換で解く

- 2 + 4 sin^{2} (x) = 0

仮定:

sin (x) = u

- 2 + 4 u^{2} = 0

- 2 + 4 u^{2} = 0 : u = \frac{1}{2}, u = - \frac{1}{2}

- 2 + 4 u^{2} = 0

2

を右側に移動します

- 2 + 4 u^{2} = 0

両辺に

2

を足す

- 2 + 4 u^{2} + 2 = 0 + 2

簡素化

4 u^{2} = 2

4 u^{2} = 2

以下で両辺を割る

4

4 u^{2} = 2

以下で両辺を割る

4

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

簡素化

u^{2} = \frac{1}{2}

u^{2} = \frac{1}{2}

x^{2} = f (a)

の場合, 解は

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

u = \frac{1}{2}, u = - \frac{1}{2}

代用を戻す

u = sin (x)

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

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

sin (x) = \frac{1}{2} : x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

sin (x) = \frac{1}{2}

以下の一般解

sin (x) = \frac{1}{2}

sin (x)

2 πn

循環を含む周期性テーブル:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

sin (x) = - \frac{1}{2} : x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

sin (x) = - \frac{1}{2}

以下の一般解

sin (x) = - \frac{1}{2}

sin (x)

2 πn

循環を含む周期性テーブル:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

すべての解を組み合わせる

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

グラフ

人気の例

2sin^2(θ)+3cos(θ)=3

2 sin^{2} (θ) + 3 cos (θ) = 3

sqrt(2)sin(θ)=1

2 sin (θ) = 1

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

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

tan(x-pi)+2sin(x-pi)=0

tan (x - π) + 2 sin (x - π) = 0

tan(45)=(1-cos(u))/(sin(u))

tan (4 5^{\circ}) = \frac{1 - cos ( u )}{sin ( u )}