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人気のある三角関数 >

2cos^2(a)tan(a)=tan(a)

解

2 cos^{2} (a) tan (a) = tan (a)

解

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

+1

度

a = 0^{\circ} + 18 0^{\circ} n, a = 13 5^{\circ} + 36 0^{\circ} n, a = 22 5^{\circ} + 36 0^{\circ} n, a = 4 5^{\circ} + 36 0^{\circ} n, a = 31 5^{\circ} + 36 0^{\circ} n

解答ステップ

2 cos^{2} (a) tan (a) = tan (a)

両辺から

tan (a)

を引く

2 cos^{2} (a) tan (a) - tan (a) = 0

因数

2 cos^{2} (a) tan (a) - tan (a) : tan (a) (2 cos (a) + 1) (2 cos (a) - 1)

2 cos^{2} (a) tan (a) - tan (a)

共通項をくくり出す

tan (a)

= tan (a) (2 cos^{2} (a) - 1)

因数

2 cos^{2} (a) - 1 : (2 cos (a) + 1) (2 cos (a) - 1)

2 cos^{2} (a) - 1

2 cos^{2} (a) - 1

を書き換え

(2 cos (a))^{2} - 1^{2}

2 cos^{2} (a) - 1

累乗根の規則を適用する:

a = (a)^{2}

2 = (2)^{2}

= (2)^{2} cos^{2} (a) - 1

1

を書き換え

1^{2}

= (2)^{2} cos^{2} (a) - 1^{2}

指数の規則を適用する:

a^{m} b^{m} = (ab)^{m}

(2)^{2} cos^{2} (a) = (2 cos (a))^{2}

= (2 cos (a))^{2} - 1^{2}

= (2 cos (a))^{2} - 1^{2}

2乗の差の公式を適用する:

x^{2} - y^{2} = (x + y) (x - y)

(2 cos (a))^{2} - 1^{2} = (2 cos (a) + 1) (2 cos (a) - 1)

= (2 cos (a) + 1) (2 cos (a) - 1)

= tan (a) (2 cos (a) + 1) (2 cos (a) - 1)

tan (a) (2 cos (a) + 1) (2 cos (a) - 1) = 0

各部分を別個に解く

tan (a) = 0 or 2 cos (a) + 1 = 0 or 2 cos (a) - 1 = 0

tan (a) = 0 : a = πn

tan (a) = 0

以下の一般解

tan (a) = 0

tan (x)

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

a = 0 + πn

a = 0 + πn

解く

a = 0 + πn : a = πn

a = 0 + πn

0 + πn = πn

a = πn

a = πn

2 cos (a) + 1 = 0 : a = \frac{3 π}{4} + 2 πn, a = \frac{5 π}{4} + 2 πn

2 cos (a) + 1 = 0

1

を右側に移動します

2 cos (a) + 1 = 0

両辺から

1

を引く

2 cos (a) + 1 - 1 = 0 - 1

簡素化

2 cos (a) = - 1

2 cos (a) = - 1

以下で両辺を割る

2

2 cos (a) = - 1

以下で両辺を割る

2

\frac{2 cos ( a )}{2} = \frac{- 1}{2}

簡素化

\frac{2 cos ( a )}{2} = \frac{- 1}{2}

簡素化

\frac{2 cos ( a )}{2} : cos (a)

\frac{2 cos ( a )}{2}

共通因数を約分する:

2

= cos (a)

簡素化

\frac{- 1}{2} : - \frac{2}{2}

\frac{- 1}{2}

分数の規則を適用する:

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

= - \frac{1}{2}

有理化する

- \frac{1}{2} : - \frac{2}{2}

- \frac{1}{2}

共役で乗じる

\frac{2}{2}

= - \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

累乗根の規則を適用する:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

cos (a) = - \frac{2}{2}

cos (a) = - \frac{2}{2}

cos (a) = - \frac{2}{2}

以下の一般解

cos (a) = - \frac{2}{2}

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

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

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

2 cos (a) - 1 = 0 : a = \frac{π}{4} + 2 πn, a = \frac{7 π}{4} + 2 πn

2 cos (a) - 1 = 0

1

を右側に移動します

2 cos (a) - 1 = 0

両辺に

1

を足す

2 cos (a) - 1 + 1 = 0 + 1

簡素化

2 cos (a) = 1

2 cos (a) = 1

以下で両辺を割る

2

2 cos (a) = 1

以下で両辺を割る

2

\frac{2 cos ( a )}{2} = \frac{1}{2}

簡素化

\frac{2 cos ( a )}{2} = \frac{1}{2}

簡素化

\frac{2 cos ( a )}{2} : cos (a)

\frac{2 cos ( a )}{2}

共通因数を約分する:

2

= cos (a)

簡素化

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

共役で乗じる

\frac{2}{2}

= \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

累乗根の規則を適用する:

a a = a

22 = 2

= 2

= \frac{2}{2}

cos (a) = \frac{2}{2}

cos (a) = \frac{2}{2}

cos (a) = \frac{2}{2}

以下の一般解

cos (a) = \frac{2}{2}

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

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

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

すべての解を組み合わせる

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

グラフ

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