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人気のある三角関数 >

tan^2(x)-sqrt(3)tan(x)=0,0<= x<= 2pi

解

tan^{2} (x) - 3 tan (x) = 0, 0 \leq x \leq 2 π

解

x = \frac{π}{3}, x = \frac{4 π}{3}, x = 0, x = π, x = 2 π

+1

度

x = 6 0^{\circ}, x = 24 0^{\circ}, x = 0^{\circ}, x = 18 0^{\circ}, x = 36 0^{\circ}

解答ステップ

tan^{2} (x) - 3 tan (x) = 0, 0 \leq x \leq 2 π

置換で解く

tan^{2} (x) - 3 tan (x) = 0

仮定:

tan (x) = u

u^{2} - 3 u = 0

u^{2} - 3 u = 0 : u = 3, u = 0

u^{2} - 3 u = 0

解くとthe二次式

u^{2} - 3 u = 0

二次Equationの公式:

次の場合:

a = 1, b = - 3, c = 0

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

(- 3)^{2} - 4 \cdot 1 \cdot 0 = 3

(- 3)^{2} - 4 \cdot 1 \cdot 0

(- 3)^{2} = 3

(- 3)^{2}

指数の規則を適用する:

n

が偶数であれば

(- a)^{n} = a^{n}

(- 3)^{2} = (3)^{2}

= (3)^{2}

累乗根の規則を適用する:

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

= (3^{\frac{1}{2}})^{2}

指数の規則を適用する:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

分数を乗じる:

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

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

共通因数を約分する:

2

= 1

= 3

4 \cdot 1 \cdot 0 = 0

4 \cdot 1 \cdot 0

規則を適用

0 \cdot a = 0

= 0

= 3 - 0

数を引く:

3 - 0 = 3

= 3

u_{1, 2} = \frac{- ( - 3 ) \pm 3}{2 \cdot 1}

解を分離する

u_{1} = \frac{- ( - 3 ) + 3}{2 \cdot 1}, u_{2} = \frac{- ( - 3 ) - 3}{2 \cdot 1}

u = \frac{- ( - 3 ) + 3}{2 \cdot 1} : 3

\frac{- ( - 3 ) + 3}{2 \cdot 1}

規則を適用

- (- a) = a

= \frac{3 + 3}{2 \cdot 1}

類似した元を足す:

3 + 3 = 23

= \frac{2 3}{2 \cdot 1}

数を乗じる:

2 \cdot 1 = 2

= \frac{2 3}{2}

数を割る:

\frac{2}{2} = 1

= 3

u = \frac{- ( - 3 ) - 3}{2 \cdot 1} : 0

\frac{- ( - 3 ) - 3}{2 \cdot 1}

規則を適用

- (- a) = a

= \frac{3 - 3}{2 \cdot 1}

類似した元を足す:

3 - 3 = 0

= \frac{0}{2 \cdot 1}

数を乗じる:

2 \cdot 1 = 2

= \frac{0}{2}

規則を適用

\frac{0}{a} = 0, a \neq = 0

= 0

二次equationの解:

u = 3, u = 0

代用を戻す

u = tan (x)

tan (x) = 3, tan (x) = 0

tan (x) = 3, tan (x) = 0

tan (x) = 3, 0 \leq x \leq 2 π : x = \frac{π}{3}, x = \frac{4 π}{3}

tan (x) = 3, 0 \leq x \leq 2 π

以下の一般解

tan (x) = 3

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}

x = \frac{π}{3} + πn

x = \frac{π}{3} + πn

範囲の解答

0 \leq x \leq 2 π

x = \frac{π}{3}, x = \frac{4 π}{3}

tan (x) = 0, 0 \leq x \leq 2 π : x = 0, x = π, x = 2 π

tan (x) = 0, 0 \leq x \leq 2 π

以下の一般解

tan (x) = 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}

x = 0 + πn

x = 0 + πn

解く

x = 0 + πn : x = πn

x = 0 + πn

0 + πn = πn

x = πn

x = πn

範囲の解答

0 \leq x \leq 2 π

x = 0, x = π, x = 2 π

すべての解を組み合わせる

x = \frac{π}{3}, x = \frac{4 π}{3}, x = 0, x = π, x = 2 π

グラフ

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