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人気のある三角関数 >

cot^2(x)-7cot(x)+10=0

解

cot^{2} (x) - 7 cot (x) + 10 = 0

解

x = 0.19739 \dots + πn, x = 0.46364 \dots + πn

+1

度

x = 11.30993 \dots^{\circ} + 18 0^{\circ} n, x = 26.56505 \dots^{\circ} + 18 0^{\circ} n

解答ステップ

cot^{2} (x) - 7 cot (x) + 10 = 0

置換で解く

cot^{2} (x) - 7 cot (x) + 10 = 0

仮定:

cot (x) = u

u^{2} - 7 u + 10 = 0

u^{2} - 7 u + 10 = 0 : u = 5, u = 2

u^{2} - 7 u + 10 = 0

解くとthe二次式

u^{2} - 7 u + 10 = 0

二次Equationの公式:

次の場合:

a = 1, b = - 7, c = 10

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 1 \cdot 10}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 1 \cdot 10}{2 \cdot 1}

(- 7)^{2} - 4 \cdot 1 \cdot 10 = 3

(- 7)^{2} - 4 \cdot 1 \cdot 10

指数の規則を適用する:

n

が偶数であれば

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

(- 7)^{2} = 7^{2}

= 7^{2} - 4 \cdot 1 \cdot 10

数を乗じる:

4 \cdot 1 \cdot 10 = 40

= 7^{2} - 40

7^{2} = 49

= 49 - 40

数を引く:

49 - 40 = 9

= 9

数を因数に分解する:

9 = 3^{2}

= 3^{2}

累乗根の規則を適用する:

n a^{n} = a

3^{2} = 3

= 3

u_{1, 2} = \frac{- ( - 7 ) \pm 3}{2 \cdot 1}

解を分離する

u_{1} = \frac{- ( - 7 ) + 3}{2 \cdot 1}, u_{2} = \frac{- ( - 7 ) - 3}{2 \cdot 1}

u = \frac{- ( - 7 ) + 3}{2 \cdot 1} : 5

\frac{- ( - 7 ) + 3}{2 \cdot 1}

規則を適用

- (- a) = a

= \frac{7 + 3}{2 \cdot 1}

数を足す:

7 + 3 = 10

= \frac{10}{2 \cdot 1}

数を乗じる:

2 \cdot 1 = 2

= \frac{10}{2}

数を割る:

\frac{10}{2} = 5

= 5

u = \frac{- ( - 7 ) - 3}{2 \cdot 1} : 2

\frac{- ( - 7 ) - 3}{2 \cdot 1}

規則を適用

- (- a) = a

= \frac{7 - 3}{2 \cdot 1}

数を引く:

7 - 3 = 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 = 5, u = 2

代用を戻す

u = cot (x)

cot (x) = 5, cot (x) = 2

cot (x) = 5, cot (x) = 2

cot (x) = 5 : x = arccot (5) + πn

cot (x) = 5

三角関数の逆数プロパティを適用する

cot (x) = 5

以下の一般解

cot (x) = 5

cot (x) = a \Rightarrow x = arccot (a) + πn

x = arccot (5) + πn

x = arccot (5) + πn

cot (x) = 2 : x = arccot (2) + πn

cot (x) = 2

三角関数の逆数プロパティを適用する

cot (x) = 2

以下の一般解

cot (x) = 2

cot (x) = a \Rightarrow x = arccot (a) + πn

x = arccot (2) + πn

x = arccot (2) + πn

すべての解を組み合わせる

x = arccot (5) + πn, x = arccot (2) + πn

10進法形式で解を証明する

x = 0.19739 \dots + πn, x = 0.46364 \dots + πn

グラフ

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