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人気のある三角関数 >

1-cos^2(θ)-6959cos(θ)=0

解

1 - cos^{2} (θ) - 6959 cos (θ) = 0

解

θ = 1.57065 \dots + 2 πn, θ = 2 π - 1.57065 \dots + 2 πn

+1

度

θ = 89.99176 \dots^{\circ} + 36 0^{\circ} n, θ = 270.00823 \dots^{\circ} + 36 0^{\circ} n

解答ステップ

1 - cos^{2} (θ) - 6959 cos (θ) = 0

置換で解く

1 - cos^{2} (θ) - 6959 cos (θ) = 0

仮定:

cos (θ) = u

1 - u^{2} - 6959 u = 0

1 - u^{2} - 6959 u = 0 : u = - \frac{6959 + 48427685}{2}, u = \frac{48427685 - 6959}{2}

1 - u^{2} - 6959 u = 0

標準的な形式で書く

a x^{2} + b x + c = 0

- u^{2} - 6959 u + 1 = 0

解くとthe二次式

- u^{2} - 6959 u + 1 = 0

二次Equationの公式:

次の場合:

a = - 1, b = - 6959, c = 1

u_{1, 2} = \frac{- ( - 6959 ) \pm ( - 6959 ) ^{2} - 4 ( - 1 ) \cdot 1}{2 ( - 1 )}

u_{1, 2} = \frac{- ( - 6959 ) \pm ( - 6959 ) ^{2} - 4 ( - 1 ) \cdot 1}{2 ( - 1 )}

(- 6959)^{2} - 4 (- 1) \cdot 1 = 48427685

(- 6959)^{2} - 4 (- 1) \cdot 1

規則を適用

- (- a) = a

= (- 6959)^{2} + 4 \cdot 1 \cdot 1

指数の規則を適用する:

n

が偶数であれば

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

(- 6959)^{2} = 695 9^{2}

= 695 9^{2} + 4 \cdot 1 \cdot 1

数を乗じる:

4 \cdot 1 \cdot 1 = 4

= 695 9^{2} + 4

695 9^{2} = 48427681

= 48427681 + 4

数を足す:

48427681 + 4 = 48427685

= 48427685

u_{1, 2} = \frac{- ( - 6959 ) \pm 48427685}{2 ( - 1 )}

解を分離する

u_{1} = \frac{- ( - 6959 ) + 48427685}{2 ( - 1 )}, u_{2} = \frac{- ( - 6959 ) - 48427685}{2 ( - 1 )}

u = \frac{- ( - 6959 ) + 48427685}{2 ( - 1 )} : - \frac{6959 + 48427685}{2}

\frac{- ( - 6959 ) + 48427685}{2 ( - 1 )}

括弧を削除する:

(- a) = - a, - (- a) = a

= \frac{6959 + 48427685}{- 2 \cdot 1}

数を乗じる:

2 \cdot 1 = 2

= \frac{6959 + 48427685}{- 2}

分数の規則を適用する:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{6959 + 48427685}{2}

u = \frac{- ( - 6959 ) - 48427685}{2 ( - 1 )} : \frac{48427685 - 6959}{2}

\frac{- ( - 6959 ) - 48427685}{2 ( - 1 )}

括弧を削除する:

(- a) = - a, - (- a) = a

= \frac{6959 - 48427685}{- 2 \cdot 1}

数を乗じる:

2 \cdot 1 = 2

= \frac{6959 - 48427685}{- 2}

分数の規則を適用する:

\frac{- a}{- b} = \frac{a}{b}

6959 - 48427685 = - (48427685 - 6959)

= \frac{48427685 - 6959}{2}

二次equationの解:

u = - \frac{6959 + 48427685}{2}, u = \frac{48427685 - 6959}{2}

代用を戻す

u = cos (θ)

cos (θ) = - \frac{6959 + 48427685}{2}, cos (θ) = \frac{48427685 - 6959}{2}

cos (θ) = - \frac{6959 + 48427685}{2}, cos (θ) = \frac{48427685 - 6959}{2}

cos (θ) = - \frac{6959 + 48427685}{2} :

解なし

cos (θ) = - \frac{6959 + 48427685}{2}

- 1 \leq cos (x) \leq 1

解なし

cos (θ) = \frac{48427685 - 6959}{2} : θ = arccos (\frac{48427685 - 6959}{2}) + 2 πn, θ = 2 π - arccos (\frac{48427685 - 6959}{2}) + 2 πn

cos (θ) = \frac{48427685 - 6959}{2}

三角関数の逆数プロパティを適用する

cos (θ) = \frac{48427685 - 6959}{2}

以下の一般解

cos (θ) = \frac{48427685 - 6959}{2}

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

θ = arccos (\frac{48427685 - 6959}{2}) + 2 πn, θ = 2 π - arccos (\frac{48427685 - 6959}{2}) + 2 πn

θ = arccos (\frac{48427685 - 6959}{2}) + 2 πn, θ = 2 π - arccos (\frac{48427685 - 6959}{2}) + 2 πn

すべての解を組み合わせる

θ = arccos (\frac{48427685 - 6959}{2}) + 2 πn, θ = 2 π - arccos (\frac{48427685 - 6959}{2}) + 2 πn

10進法形式で解を証明する

θ = 1.57065 \dots + 2 πn, θ = 2 π - 1.57065 \dots + 2 πn

グラフ

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