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人気のある三角関数 >

2cos^2(t)-7cos(t)+5=0

解

2 cos^{2} (t) - 7 cos (t) + 5 = 0

解

t = 2 πn

+1

度

t = 0^{\circ} + 36 0^{\circ} n

解答ステップ

2 cos^{2} (t) - 7 cos (t) + 5 = 0

置換で解く

2 cos^{2} (t) - 7 cos (t) + 5 = 0

仮定:

cos (t) = u

2 u^{2} - 7 u + 5 = 0

2 u^{2} - 7 u + 5 = 0 : u = \frac{5}{2}, u = 1

2 u^{2} - 7 u + 5 = 0

解くとthe二次式

2 u^{2} - 7 u + 5 = 0

二次Equationの公式:

次の場合:

a = 2, b = - 7, c = 5

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 2 \cdot 5}{2 \cdot 2}

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 2 \cdot 5}{2 \cdot 2}

(- 7)^{2} - 4 \cdot 2 \cdot 5 = 3

(- 7)^{2} - 4 \cdot 2 \cdot 5

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 2 \cdot 5

数を乗じる:

4 \cdot 2 \cdot 5 = 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 2}

解を分離する

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

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

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

規則を適用

- (- a) = a

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

数を足す:

7 + 3 = 10

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

数を乗じる:

2 \cdot 2 = 4

= \frac{10}{4}

共通因数を約分する:

2

= \frac{5}{2}

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

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

規則を適用

- (- a) = a

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

数を引く:

7 - 3 = 4

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

数を乗じる:

2 \cdot 2 = 4

= \frac{4}{4}

規則を適用

\frac{a}{a} = 1

= 1

二次equationの解:

u = \frac{5}{2}, u = 1

代用を戻す

u = cos (t)

cos (t) = \frac{5}{2}, cos (t) = 1

cos (t) = \frac{5}{2}, cos (t) = 1

cos (t) = \frac{5}{2} :

解なし

cos (t) = \frac{5}{2}

- 1 \leq cos (x) \leq 1

解なし

cos (t) = 1 : t = 2 πn

cos (t) = 1

以下の一般解

cos (t) = 1

cos (x)

2 πn

循環を含む周期性テーブル:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

t = 0 + 2 πn

t = 0 + 2 πn

解く

t = 0 + 2 πn : t = 2 πn

t = 0 + 2 πn

0 + 2 πn = 2 πn

t = 2 πn

t = 2 πn

すべての解を組み合わせる

t = 2 πn

グラフ

人気の例

(cos (x)) = 0

arccos (x) = \frac{3}{8}

sqrt(2)sin(x)+1=cos(2x)

2 sin (x) + 1 = cos (2 x)

sin(2x)=-1sin(x)

sin (2 x) = - 1 sin (x)

cos(x)= 1/(sqrt(6))

cos (x) = \frac{1}{6}