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人気のある三角関数 >

solvefor θ,cos(θ)+cos(90-θ)=0

解

解く

θ, cos (θ) + cos (9 0^{\circ} - θ) = 0

解

θ = 13 5^{\circ} + 36 0^{\circ} n, θ = 31 5^{\circ} + 36 0^{\circ} n

+1

ラジアン

θ = \frac{3 π}{4} + 2 πn, θ = \frac{7 π}{4} + 2 πn

解答ステップ

cos (θ) + cos (9 0^{\circ} - θ) = 0

三角関数の公式を使用して書き換える

cos (θ) + cos (9 0^{\circ} - θ)

和・積の公式を使用する:

cos (s) + cos (t) = 2 cos (\frac{s + t}{2}) cos (\frac{s - t}{2})

= 2 cos (\frac{θ + 9 0 ^{\circ} - θ}{2}) cos (\frac{θ - ( 9 0 ^{\circ} - θ )}{2})

簡素化

2 cos (\frac{θ + 9 0 ^{\circ} - θ}{2}) cos (\frac{θ - ( 9 0 ^{\circ} - θ )}{2}) : 2 cos (\frac{4 θ - 18 0 ^{\circ}}{4})

2 cos (\frac{θ + 9 0 ^{\circ} - θ}{2}) cos (\frac{θ - ( 9 0 ^{\circ} - θ )}{2})

\frac{θ + 9 0 ^{\circ} - θ}{2} = 4 5^{\circ}

\frac{θ + 9 0 ^{\circ} - θ}{2}

θ + 9 0^{\circ} - θ = 9 0^{\circ}

θ + 9 0^{\circ} - θ

条件のようなグループ

= θ - θ + 9 0^{\circ}

類似した元を足す:

θ - θ = 0

= 9 0^{\circ}

= \frac{9 0 ^{\circ}}{2}

分数の規則を適用する:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{18 0 ^{\circ}}{2 \cdot 2}

数を乗じる:

2 \cdot 2 = 4

= 4 5^{\circ}

= 2 cos (4 5^{\circ}) cos (\frac{θ - ( - θ + 9 0 ^{\circ} )}{2})

\frac{θ - ( 9 0 ^{\circ} - θ )}{2} = \frac{4 θ - 18 0 ^{\circ}}{4}

\frac{θ - ( 9 0 ^{\circ} - θ )}{2}

拡張

θ - (9 0^{\circ} - θ) : 2 θ - 9 0^{\circ}

θ - (9 0^{\circ} - θ)

- (9 0^{\circ} - θ) : - 9 0^{\circ} + θ

- (9 0^{\circ} - θ)

括弧を分配する

= - (9 0^{\circ}) - (- θ)

マイナス・プラスの規則を適用する

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

= - 9 0^{\circ} + θ

= θ - 9 0^{\circ} + θ

簡素化

θ - 9 0^{\circ} + θ : 2 θ - 9 0^{\circ}

θ - 9 0^{\circ} + θ

条件のようなグループ

= θ + θ - 9 0^{\circ}

類似した元を足す:

θ + θ = 2 θ

= 2 θ - 9 0^{\circ}

= 2 θ - 9 0^{\circ}

= \frac{2 θ - 9 0 ^{\circ}}{2}

結合

2 θ - 9 0^{\circ} : \frac{4 θ - 18 0 ^{\circ}}{2}

2 θ - 9 0^{\circ}

元を分数に変換する:

2 θ = \frac{2 θ 2}{2}

= \frac{2 θ \cdot 2}{2} - 9 0^{\circ}

分母が等しいので, 分数を組み合わせる:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{2 θ \cdot 2 - 18 0 ^{\circ}}{2}

数を乗じる:

2 \cdot 2 = 4

= \frac{4 θ - 18 0 ^{\circ}}{2}

= \frac{\frac{4 θ - 18 0 ^{\circ}}{2}}{2}

分数の規則を適用する:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{4 θ - 18 0 ^{\circ}}{2 \cdot 2}

数を乗じる:

2 \cdot 2 = 4

= \frac{4 θ - 18 0 ^{\circ}}{4}

= 2 cos (4 5^{\circ}) cos (\frac{4 θ - 18 0 ^{\circ}}{4})

簡素化

cos (4 5^{\circ}) : \frac{2}{2}

cos (4 5^{\circ})

次の自明恒等式を使用する:

cos (4 5^{\circ}) = \frac{2}{2}

cos (x)

36 0^{\circ} n

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}

= \frac{2}{2}

= 2 \cdot \frac{2}{2} cos (\frac{4 θ - 18 0 ^{\circ}}{4})

分数を乗じる:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 \cdot 2 cos ( \frac{4 θ - 18 0 ^{\circ}}{4} )}{2}

共通因数を約分する:

2

= 2 cos (\frac{4 θ - 18 0 ^{\circ}}{4})

= 2 cos (\frac{4 θ - 18 0 ^{\circ}}{4})

2 cos (\frac{4 θ - 18 0 ^{\circ}}{4}) = 0

以下で両辺を割る

2

2 cos (\frac{4 θ - 18 0 ^{\circ}}{4}) = 0

以下で両辺を割る

2

\frac{2 cos ( \frac{4 θ - 18 0 ^{\circ}}{4} )}{2} = \frac{0}{2}

簡素化

cos (\frac{4 θ - 18 0 ^{\circ}}{4}) = 0

cos (\frac{4 θ - 18 0 ^{\circ}}{4}) = 0

以下の一般解

cos (\frac{4 θ - 18 0 ^{\circ}}{4}) = 0

cos (x)

36 0^{\circ} n

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}

\frac{4 θ - 18 0 ^{\circ}}{4} = 9 0^{\circ} + 36 0^{\circ} n, \frac{4 θ - 18 0 ^{\circ}}{4} = 27 0^{\circ} + 36 0^{\circ} n

\frac{4 θ - 18 0 ^{\circ}}{4} = 9 0^{\circ} + 36 0^{\circ} n, \frac{4 θ - 18 0 ^{\circ}}{4} = 27 0^{\circ} + 36 0^{\circ} n

解く

\frac{4 θ - 18 0 ^{\circ}}{4} = 9 0^{\circ} + 36 0^{\circ} n : θ = 13 5^{\circ} + 36 0^{\circ} n

\frac{4 θ - 18 0 ^{\circ}}{4} = 9 0^{\circ} + 36 0^{\circ} n

以下で両辺を乗じる:

4

\frac{4 θ - 18 0 ^{\circ}}{4} = 9 0^{\circ} + 36 0^{\circ} n

以下で両辺を乗じる:

4

\frac{4 ( 4 θ - 18 0 ^{\circ} )}{4} = 4 \cdot 9 0^{\circ} + 4 \cdot 36 0^{\circ} n

簡素化

\frac{4 ( 4 θ - 18 0 ^{\circ} )}{4} = 4 \cdot 9 0^{\circ} + 4 \cdot 36 0^{\circ} n

簡素化

\frac{4 ( 4 θ - 18 0 ^{\circ} )}{4} : 4 θ - 18 0^{\circ}

\frac{4 ( 4 θ - 18 0 ^{\circ} )}{4}

数を割る:

\frac{4}{4} = 1

= 4 θ - 18 0^{\circ}

簡素化

4 \cdot 9 0^{\circ} + 4 \cdot 36 0^{\circ} n : 36 0^{\circ} + 144 0^{\circ} n

4 \cdot 9 0^{\circ} + 4 \cdot 36 0^{\circ} n

4 \cdot 9 0^{\circ} = 36 0^{\circ}

4 \cdot 9 0^{\circ}

分数を乗じる:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= 36 0^{\circ}

数を割る:

\frac{4}{2} = 2

= 36 0^{\circ}

4 \cdot 36 0^{\circ} n = 144 0^{\circ} n

4 \cdot 36 0^{\circ} n

数を乗じる:

4 \cdot 2 = 8

= 144 0^{\circ} n

= 36 0^{\circ} + 144 0^{\circ} n

4 θ - 18 0^{\circ} = 36 0^{\circ} + 144 0^{\circ} n

4 θ - 18 0^{\circ} = 36 0^{\circ} + 144 0^{\circ} n

4 θ - 18 0^{\circ} = 36 0^{\circ} + 144 0^{\circ} n

18 0^{\circ}

を右側に移動します

4 θ - 18 0^{\circ} = 36 0^{\circ} + 144 0^{\circ} n

両辺に

18 0^{\circ}

を足す

4 θ - 18 0^{\circ} + 18 0^{\circ} = 36 0^{\circ} + 144 0^{\circ} n + 18 0^{\circ}

簡素化

4 θ = 54 0^{\circ} + 144 0^{\circ} n

4 θ = 54 0^{\circ} + 144 0^{\circ} n

以下で両辺を割る

4

4 θ = 54 0^{\circ} + 144 0^{\circ} n

以下で両辺を割る

4

\frac{4 θ}{4} = 13 5^{\circ} + \frac{144 0 ^{\circ} n}{4}

簡素化

θ = 13 5^{\circ} + 36 0^{\circ} n

θ = 13 5^{\circ} + 36 0^{\circ} n

解く

\frac{4 θ - 18 0 ^{\circ}}{4} = 27 0^{\circ} + 36 0^{\circ} n : θ = 31 5^{\circ} + 36 0^{\circ} n

\frac{4 θ - 18 0 ^{\circ}}{4} = 27 0^{\circ} + 36 0^{\circ} n

以下で両辺を乗じる:

4

\frac{4 θ - 18 0 ^{\circ}}{4} = 27 0^{\circ} + 36 0^{\circ} n

以下で両辺を乗じる:

4

\frac{4 ( 4 θ - 18 0 ^{\circ} )}{4} = 4 \cdot 27 0^{\circ} + 4 \cdot 36 0^{\circ} n

簡素化

\frac{4 ( 4 θ - 18 0 ^{\circ} )}{4} = 4 \cdot 27 0^{\circ} + 4 \cdot 36 0^{\circ} n

簡素化

\frac{4 ( 4 θ - 18 0 ^{\circ} )}{4} : 4 θ - 18 0^{\circ}

\frac{4 ( 4 θ - 18 0 ^{\circ} )}{4}

数を割る:

\frac{4}{4} = 1

= 4 θ - 18 0^{\circ}

簡素化

4 \cdot 27 0^{\circ} + 4 \cdot 36 0^{\circ} n : 108 0^{\circ} + 144 0^{\circ} n

4 \cdot 27 0^{\circ} + 4 \cdot 36 0^{\circ} n

4 \cdot 27 0^{\circ} = 108 0^{\circ}

4 \cdot 27 0^{\circ}

分数を乗じる:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= 108 0^{\circ}

数を乗じる:

3 \cdot 4 = 12

= 108 0^{\circ}

数を割る:

\frac{12}{2} = 6

= 108 0^{\circ}

4 \cdot 36 0^{\circ} n = 144 0^{\circ} n

4 \cdot 36 0^{\circ} n

数を乗じる:

4 \cdot 2 = 8

= 144 0^{\circ} n

= 108 0^{\circ} + 144 0^{\circ} n

4 θ - 18 0^{\circ} = 108 0^{\circ} + 144 0^{\circ} n

4 θ - 18 0^{\circ} = 108 0^{\circ} + 144 0^{\circ} n

4 θ - 18 0^{\circ} = 108 0^{\circ} + 144 0^{\circ} n

18 0^{\circ}

を右側に移動します

4 θ - 18 0^{\circ} = 108 0^{\circ} + 144 0^{\circ} n

両辺に

18 0^{\circ}

を足す

4 θ - 18 0^{\circ} + 18 0^{\circ} = 108 0^{\circ} + 144 0^{\circ} n + 18 0^{\circ}

簡素化

4 θ = 126 0^{\circ} + 144 0^{\circ} n

4 θ = 126 0^{\circ} + 144 0^{\circ} n

以下で両辺を割る

4

4 θ = 126 0^{\circ} + 144 0^{\circ} n

以下で両辺を割る

4

\frac{4 θ}{4} = 31 5^{\circ} + \frac{144 0 ^{\circ} n}{4}

簡素化

θ = 31 5^{\circ} + 36 0^{\circ} n

θ = 31 5^{\circ} + 36 0^{\circ} n

θ = 13 5^{\circ} + 36 0^{\circ} n, θ = 31 5^{\circ} + 36 0^{\circ} n

グラフ

人気の例

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