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人気のある三角関数 >

75cos^2(x)-34cos(x)+3=0

解

75 cos^{2} (x) - 34 cos (x) + 3 = 0

解

x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn, x = 1.45050 \dots + 2 πn, x = 2 π - 1.45050 \dots + 2 πn

+1

度

x = 70.52877 \dots^{\circ} + 36 0^{\circ} n, x = 289.47122 \dots^{\circ} + 36 0^{\circ} n, x = 83.10789 \dots^{\circ} + 36 0^{\circ} n, x = 276.89210 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

75 cos^{2} (x) - 34 cos (x) + 3 = 0

置換で解く

75 cos^{2} (x) - 34 cos (x) + 3 = 0

仮定:

cos (x) = u

75 u^{2} - 34 u + 3 = 0

75 u^{2} - 34 u + 3 = 0 : u = \frac{1}{3}, u = \frac{3}{25}

75 u^{2} - 34 u + 3 = 0

解くとthe二次式

75 u^{2} - 34 u + 3 = 0

二次Equationの公式:

次の場合:

a = 75, b = - 34, c = 3

u_{1, 2} = \frac{- ( - 34 ) \pm ( - 34 ) ^{2} - 4 \cdot 75 \cdot 3}{2 \cdot 75}

u_{1, 2} = \frac{- ( - 34 ) \pm ( - 34 ) ^{2} - 4 \cdot 75 \cdot 3}{2 \cdot 75}

(- 34)^{2} - 4 \cdot 75 \cdot 3 = 16

(- 34)^{2} - 4 \cdot 75 \cdot 3

指数の規則を適用する:

n

が偶数であれば

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

(- 34)^{2} = 3 4^{2}

= 3 4^{2} - 4 \cdot 75 \cdot 3

数を乗じる:

4 \cdot 75 \cdot 3 = 900

= 3 4^{2} - 900

3 4^{2} = 1156

= 1156 - 900

数を引く:

1156 - 900 = 256

= 256

数を因数に分解する:

256 = 1 6^{2}

= 1 6^{2}

累乗根の規則を適用する:

n a^{n} = a

1 6^{2} = 16

= 16

u_{1, 2} = \frac{- ( - 34 ) \pm 16}{2 \cdot 75}

解を分離する

u_{1} = \frac{- ( - 34 ) + 16}{2 \cdot 75}, u_{2} = \frac{- ( - 34 ) - 16}{2 \cdot 75}

u = \frac{- ( - 34 ) + 16}{2 \cdot 75} : \frac{1}{3}

\frac{- ( - 34 ) + 16}{2 \cdot 75}

規則を適用

- (- a) = a

= \frac{34 + 16}{2 \cdot 75}

数を足す:

34 + 16 = 50

= \frac{50}{2 \cdot 75}

数を乗じる:

2 \cdot 75 = 150

= \frac{50}{150}

共通因数を約分する:

50

= \frac{1}{3}

u = \frac{- ( - 34 ) - 16}{2 \cdot 75} : \frac{3}{25}

\frac{- ( - 34 ) - 16}{2 \cdot 75}

規則を適用

- (- a) = a

= \frac{34 - 16}{2 \cdot 75}

数を引く:

34 - 16 = 18

= \frac{18}{2 \cdot 75}

数を乗じる:

2 \cdot 75 = 150

= \frac{18}{150}

共通因数を約分する:

6

= \frac{3}{25}

二次equationの解:

u = \frac{1}{3}, u = \frac{3}{25}

代用を戻す

u = cos (x)

cos (x) = \frac{1}{3}, cos (x) = \frac{3}{25}

cos (x) = \frac{1}{3}, cos (x) = \frac{3}{25}

cos (x) = \frac{1}{3} : x = arccos (\frac{1}{3}) + 2 πn, x = 2 π - arccos (\frac{1}{3}) + 2 πn

cos (x) = \frac{1}{3}

三角関数の逆数プロパティを適用する

cos (x) = \frac{1}{3}

以下の一般解

cos (x) = \frac{1}{3}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{1}{3}) + 2 πn, x = 2 π - arccos (\frac{1}{3}) + 2 πn

x = arccos (\frac{1}{3}) + 2 πn, x = 2 π - arccos (\frac{1}{3}) + 2 πn

cos (x) = \frac{3}{25} : x = arccos (\frac{3}{25}) + 2 πn, x = 2 π - arccos (\frac{3}{25}) + 2 πn

cos (x) = \frac{3}{25}

三角関数の逆数プロパティを適用する

cos (x) = \frac{3}{25}

以下の一般解

cos (x) = \frac{3}{25}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{3}{25}) + 2 πn, x = 2 π - arccos (\frac{3}{25}) + 2 πn

x = arccos (\frac{3}{25}) + 2 πn, x = 2 π - arccos (\frac{3}{25}) + 2 πn

すべての解を組み合わせる

x = arccos (\frac{1}{3}) + 2 πn, x = 2 π - arccos (\frac{1}{3}) + 2 πn, x = arccos (\frac{3}{25}) + 2 πn, x = 2 π - arccos (\frac{3}{25}) + 2 πn

10進法形式で解を証明する

x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn, x = 1.45050 \dots + 2 πn, x = 2 π - 1.45050 \dots + 2 πn

グラフ

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