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人気のある三角関数 >

tan^2(x)-5tan(x)-9=0

解

tan^{2} (x) - 5 tan (x) - 9 = 0

解

x = 1.41592 \dots + πn, x = - 0.95227 \dots + πn

+1

度

x = 81.12633 \dots^{\circ} + 18 0^{\circ} n, x = - 54.56128 \dots^{\circ} + 18 0^{\circ} n

解答ステップ

tan^{2} (x) - 5 tan (x) - 9 = 0

置換で解く

tan^{2} (x) - 5 tan (x) - 9 = 0

仮定:

tan (x) = u

u^{2} - 5 u - 9 = 0

u^{2} - 5 u - 9 = 0 : u = \frac{5 + 61}{2}, u = \frac{5 - 61}{2}

u^{2} - 5 u - 9 = 0

解くとthe二次式

u^{2} - 5 u - 9 = 0

二次Equationの公式:

次の場合:

a = 1, b = - 5, c = - 9

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 1 \cdot ( - 9 )}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 1 \cdot ( - 9 )}{2 \cdot 1}

(- 5)^{2} - 4 \cdot 1 \cdot (- 9) = 61

(- 5)^{2} - 4 \cdot 1 \cdot (- 9)

規則を適用

- (- a) = a

= (- 5)^{2} + 4 \cdot 1 \cdot 9

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 9

数を乗じる:

4 \cdot 1 \cdot 9 = 36

= 5^{2} + 36

5^{2} = 25

= 25 + 36

数を足す:

25 + 36 = 61

= 61

u_{1, 2} = \frac{- ( - 5 ) \pm 61}{2 \cdot 1}

解を分離する

u_{1} = \frac{- ( - 5 ) + 61}{2 \cdot 1}, u_{2} = \frac{- ( - 5 ) - 61}{2 \cdot 1}

u = \frac{- ( - 5 ) + 61}{2 \cdot 1} : \frac{5 + 61}{2}

\frac{- ( - 5 ) + 61}{2 \cdot 1}

規則を適用

- (- a) = a

= \frac{5 + 61}{2 \cdot 1}

数を乗じる:

2 \cdot 1 = 2

= \frac{5 + 61}{2}

u = \frac{- ( - 5 ) - 61}{2 \cdot 1} : \frac{5 - 61}{2}

\frac{- ( - 5 ) - 61}{2 \cdot 1}

規則を適用

- (- a) = a

= \frac{5 - 61}{2 \cdot 1}

数を乗じる:

2 \cdot 1 = 2

= \frac{5 - 61}{2}

二次equationの解:

u = \frac{5 + 61}{2}, u = \frac{5 - 61}{2}

代用を戻す

u = tan (x)

tan (x) = \frac{5 + 61}{2}, tan (x) = \frac{5 - 61}{2}

tan (x) = \frac{5 + 61}{2}, tan (x) = \frac{5 - 61}{2}

tan (x) = \frac{5 + 61}{2} : x = arctan (\frac{5 + 61}{2}) + πn

tan (x) = \frac{5 + 61}{2}

三角関数の逆数プロパティを適用する

tan (x) = \frac{5 + 61}{2}

以下の一般解

tan (x) = \frac{5 + 61}{2}

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

x = arctan (\frac{5 + 61}{2}) + πn

x = arctan (\frac{5 + 61}{2}) + πn

tan (x) = \frac{5 - 61}{2} : x = arctan (\frac{5 - 61}{2}) + πn

tan (x) = \frac{5 - 61}{2}

三角関数の逆数プロパティを適用する

tan (x) = \frac{5 - 61}{2}

以下の一般解

tan (x) = \frac{5 - 61}{2}

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

x = arctan (\frac{5 - 61}{2}) + πn

x = arctan (\frac{5 - 61}{2}) + πn

すべての解を組み合わせる

x = arctan (\frac{5 + 61}{2}) + πn, x = arctan (\frac{5 - 61}{2}) + πn

10進法形式で解を証明する

x = 1.41592 \dots + πn, x = - 0.95227 \dots + πn

グラフ

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