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人気のある三角関数 >

tan^2(x)-5tan(x)+6=0

解

tan^{2} (x) - 5 tan (x) + 6 = 0

解

x = 1.24904 \dots + πn, x = 1.10714 \dots + πn

+1

度

x = 71.56505 \dots^{\circ} + 18 0^{\circ} n, x = 63.43494 \dots^{\circ} + 18 0^{\circ} n

解答ステップ

tan^{2} (x) - 5 tan (x) + 6 = 0

置換で解く

tan^{2} (x) - 5 tan (x) + 6 = 0

仮定:

tan (x) = u

u^{2} - 5 u + 6 = 0

u^{2} - 5 u + 6 = 0 : u = 3, u = 2

u^{2} - 5 u + 6 = 0

解くとthe二次式

u^{2} - 5 u + 6 = 0

二次Equationの公式:

次の場合:

a = 1, b = - 5, c = 6

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 1 \cdot 6}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 1 \cdot 6}{2 \cdot 1}

(- 5)^{2} - 4 \cdot 1 \cdot 6 = 1

(- 5)^{2} - 4 \cdot 1 \cdot 6

指数の規則を適用する:

n

が偶数であれば

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

(- 5)^{2} = 5^{2}

= 5^{2} - 4 \cdot 1 \cdot 6

数を乗じる:

4 \cdot 1 \cdot 6 = 24

= 5^{2} - 24

5^{2} = 25

= 25 - 24

数を引く:

25 - 24 = 1

= 1

規則を適用

1 = 1

= 1

u_{1, 2} = \frac{- ( - 5 ) \pm 1}{2 \cdot 1}

解を分離する

u_{1} = \frac{- ( - 5 ) + 1}{2 \cdot 1}, u_{2} = \frac{- ( - 5 ) - 1}{2 \cdot 1}

u = \frac{- ( - 5 ) + 1}{2 \cdot 1} : 3

\frac{- ( - 5 ) + 1}{2 \cdot 1}

規則を適用

- (- a) = a

= \frac{5 + 1}{2 \cdot 1}

数を足す:

5 + 1 = 6

= \frac{6}{2 \cdot 1}

数を乗じる:

2 \cdot 1 = 2

= \frac{6}{2}

数を割る:

\frac{6}{2} = 3

= 3

u = \frac{- ( - 5 ) - 1}{2 \cdot 1} : 2

\frac{- ( - 5 ) - 1}{2 \cdot 1}

規則を適用

- (- a) = a

= \frac{5 - 1}{2 \cdot 1}

数を引く:

5 - 1 = 4

= \frac{4}{2 \cdot 1}

数を乗じる:

2 \cdot 1 = 2

= \frac{4}{2}

数を割る:

\frac{4}{2} = 2

= 2

二次equationの解:

u = 3, u = 2

代用を戻す

u = tan (x)

tan (x) = 3, tan (x) = 2

tan (x) = 3, tan (x) = 2

tan (x) = 3 : x = arctan (3) + πn

tan (x) = 3

三角関数の逆数プロパティを適用する

tan (x) = 3

以下の一般解

tan (x) = 3

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (3) + πn

x = arctan (3) + πn

tan (x) = 2 : x = arctan (2) + πn

tan (x) = 2

三角関数の逆数プロパティを適用する

tan (x) = 2

以下の一般解

tan (x) = 2

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (2) + πn

x = arctan (2) + πn

すべての解を組み合わせる

x = arctan (3) + πn, x = arctan (2) + πn

10進法形式で解を証明する

x = 1.24904 \dots + πn, x = 1.10714 \dots + πn

グラフ

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