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人気のある三角関数 >

証明する (cot(3θ)-tan(3θ))/(sin(3θ)+cos(3θ))=csc(3θ)-sec(3θ)

解

証明する

\frac{cot ( 3 θ ) - tan ( 3 θ )}{sin ( 3 θ ) + cos ( 3 θ )} = csc (3 θ) - sec (3 θ)

解

真

解答ステップ

\frac{cot ( 3 θ ) - tan ( 3 θ )}{sin ( 3 θ ) + cos ( 3 θ )} = csc (3 θ) - sec (3 θ)

左側を操作する

\frac{cot ( 3 θ ) - tan ( 3 θ )}{sin ( 3 θ ) + cos ( 3 θ )}

サイン, コサインで表わす

\frac{cot ( 3 θ ) - tan ( 3 θ )}{cos ( 3 θ ) + sin ( 3 θ )}

基本的な三角関数の公式を使用する:

cot (x) = \frac{cos ( x )}{sin ( x )}

= \frac{\frac{c o s ( 3 θ )}{s i n ( 3 θ )} - tan ( 3 θ )}{cos ( 3 θ ) + sin ( 3 θ )}

基本的な三角関数の公式を使用する:

tan (x) = \frac{sin ( x )}{cos ( x )}

= \frac{\frac{c o s ( 3 θ )}{s i n ( 3 θ )} - \frac{s i n ( 3 θ )}{c o s ( 3 θ )}}{cos ( 3 θ ) + sin ( 3 θ )}

簡素化

\frac{\frac{c o s ( 3 θ )}{s i n ( 3 θ )} - \frac{s i n ( 3 θ )}{c o s ( 3 θ )}}{cos ( 3 θ ) + sin ( 3 θ )} : \frac{cos ( 3 θ ) - sin ( 3 θ )}{sin ( 3 θ ) cos ( 3 θ )}

\frac{\frac{c o s ( 3 θ )}{s i n ( 3 θ )} - \frac{s i n ( 3 θ )}{c o s ( 3 θ )}}{cos ( 3 θ ) + sin ( 3 θ )}

結合

\frac{cos ( 3 θ )}{sin ( 3 θ )} - \frac{sin ( 3 θ )}{cos ( 3 θ )} : \frac{cos ^{2} ( 3 θ ) - sin ^{2} ( 3 θ )}{sin ( 3 θ ) cos ( 3 θ )}

\frac{cos ( 3 θ )}{sin ( 3 θ )} - \frac{sin ( 3 θ )}{cos ( 3 θ )}

以下の最小公倍数:

sin (3 θ), cos (3 θ) : sin (3 θ) cos (3 θ)

sin (3 θ), cos (3 θ)

最小公倍数 (LCM)

sin (3 θ)

または以下のいずれかに現れる因数で構成された式を計算する:

cos (3 θ)

= sin (3 θ) cos (3 θ)

LCMに基づいて分数を調整する

該当する分母を乗じてLCMに変えるために

必要な量で各分子を乗じる

sin (3 θ) cos (3 θ)

\frac{cos ( 3 θ )}{sin ( 3 θ )}

の場合

:

分母と分子に以下を乗じる:

cos (3 θ)

\frac{cos ( 3 θ )}{sin ( 3 θ )} = \frac{cos ( 3 θ ) cos ( 3 θ )}{sin ( 3 θ ) cos ( 3 θ )} = \frac{cos ^{2} ( 3 θ )}{sin ( 3 θ ) cos ( 3 θ )}

\frac{sin ( 3 θ )}{cos ( 3 θ )}

の場合

:

分母と分子に以下を乗じる:

sin (3 θ)

\frac{sin ( 3 θ )}{cos ( 3 θ )} = \frac{sin ( 3 θ ) sin ( 3 θ )}{cos ( 3 θ ) sin ( 3 θ )} = \frac{sin ^{2} ( 3 θ )}{sin ( 3 θ ) cos ( 3 θ )}

= \frac{cos ^{2} ( 3 θ )}{sin ( 3 θ ) cos ( 3 θ )} - \frac{sin ^{2} ( 3 θ )}{sin ( 3 θ ) cos ( 3 θ )}

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{cos ^{2} ( 3 θ ) - sin ^{2} ( 3 θ )}{sin ( 3 θ ) cos ( 3 θ )}

= \frac{\frac{c o s ^{2} ( 3 θ ) - s i n ^{2} ( 3 θ )}{s i n ( 3 θ ) c o s ( 3 θ )}}{cos ( 3 θ ) + sin ( 3 θ )}

分数の規則を適用する:

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

= \frac{cos ^{2} ( 3 θ ) - sin ^{2} ( 3 θ )}{sin ( 3 θ ) cos ( 3 θ ) ( cos ( 3 θ ) + sin ( 3 θ ) )}

2乗の差の公式を適用する:

x^{2} - y^{2} = (x + y) (x - y)

cos^{2} (3 θ) - sin^{2} (3 θ) = (cos (3 θ) + sin (3 θ)) (cos (3 θ) - sin (3 θ))

= \frac{( cos ( 3 θ ) + sin ( 3 θ ) ) ( cos ( 3 θ ) - sin ( 3 θ ) )}{sin ( 3 θ ) cos ( 3 θ ) ( cos ( 3 θ ) + sin ( 3 θ ) )}

共通因数を約分する:

cos (3 θ) + sin (3 θ)

= \frac{cos ( 3 θ ) - sin ( 3 θ )}{sin ( 3 θ ) cos ( 3 θ )}

= \frac{cos ( 3 θ ) - sin ( 3 θ )}{sin ( 3 θ ) cos ( 3 θ )}

= \frac{cos ( 3 θ ) - sin ( 3 θ )}{cos ( 3 θ ) sin ( 3 θ )}

三角関数の公式を使用して書き換える

基本的な三角関数の公式を使用する:

sin (x) = \frac{1}{csc ( x )}

\frac{cos ( 3 θ ) - \frac{1}{c s c ( 3 θ )}}{cos ( 3 θ ) \frac{1}{c s c ( 3 θ )}}

基本的な三角関数の公式を使用する:

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

\frac{\frac{1}{s e c ( 3 θ )} - \frac{1}{c s c ( 3 θ )}}{\frac{1}{s e c ( 3 θ )} \cdot \frac{1}{c s c ( 3 θ )}}

簡素化

\frac{\frac{1}{s e c ( 3 θ )} - \frac{1}{c s c ( 3 θ )}}{\frac{1}{s e c ( 3 θ )} \cdot \frac{1}{c s c ( 3 θ )}}

乗じる

\frac{1}{sec ( 3 θ )} \cdot \frac{1}{csc ( 3 θ )} : \frac{1}{sec ( 3 θ ) csc ( 3 θ )}

\frac{1}{sec ( 3 θ )} \cdot \frac{1}{csc ( 3 θ )}

分数を乗じる:

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

= \frac{1 \cdot 1}{sec ( 3 θ ) csc ( 3 θ )}

数を乗じる:

1 \cdot 1 = 1

= \frac{1}{sec ( 3 θ ) csc ( 3 θ )}

= \frac{\frac{1}{s e c ( 3 θ )} - \frac{1}{c s c ( 3 θ )}}{\frac{1}{s e c ( 3 θ ) c s c ( 3 θ )}}

結合

\frac{1}{sec ( 3 θ )} - \frac{1}{csc ( 3 θ )} : \frac{csc ( 3 θ ) - sec ( 3 θ )}{sec ( 3 θ ) csc ( 3 θ )}

\frac{1}{sec ( 3 θ )} - \frac{1}{csc ( 3 θ )}

以下の最小公倍数:

sec (3 θ), csc (3 θ) : sec (3 θ) csc (3 θ)

sec (3 θ), csc (3 θ)

最小公倍数 (LCM)

sec (3 θ)

または以下のいずれかに現れる因数で構成された式を計算する:

csc (3 θ)

= sec (3 θ) csc (3 θ)

LCMに基づいて分数を調整する

該当する分母を乗じてLCMに変えるために

必要な量で各分子を乗じる

sec (3 θ) csc (3 θ)

\frac{1}{sec ( 3 θ )}

の場合

:

分母と分子に以下を乗じる:

csc (3 θ)

\frac{1}{sec ( 3 θ )} = \frac{1 \cdot csc ( 3 θ )}{sec ( 3 θ ) csc ( 3 θ )} = \frac{csc ( 3 θ )}{sec ( 3 θ ) csc ( 3 θ )}

\frac{1}{csc ( 3 θ )}

の場合

:

分母と分子に以下を乗じる:

sec (3 θ)

\frac{1}{csc ( 3 θ )} = \frac{1 \cdot sec ( 3 θ )}{csc ( 3 θ ) sec ( 3 θ )} = \frac{sec ( 3 θ )}{sec ( 3 θ ) csc ( 3 θ )}

= \frac{csc ( 3 θ )}{sec ( 3 θ ) csc ( 3 θ )} - \frac{sec ( 3 θ )}{sec ( 3 θ ) csc ( 3 θ )}

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{csc ( 3 θ ) - sec ( 3 θ )}{sec ( 3 θ ) csc ( 3 θ )}

= \frac{\frac{c s c ( 3 θ ) - s e c ( 3 θ )}{s e c ( 3 θ ) c s c ( 3 θ )}}{\frac{1}{s e c ( 3 θ ) c s c ( 3 θ )}}

分数を割る:

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

= \frac{( csc ( 3 θ ) - sec ( 3 θ ) ) sec ( 3 θ ) csc ( 3 θ )}{sec ( 3 θ ) csc ( 3 θ ) \cdot 1}

改良

= \frac{( csc ( 3 θ ) - sec ( 3 θ ) ) sec ( 3 θ ) csc ( 3 θ )}{sec ( 3 θ ) csc ( 3 θ )}

共通因数を約分する:

sec (3 θ)

= \frac{( csc ( 3 θ ) - sec ( 3 θ ) ) csc ( 3 θ )}{csc ( 3 θ )}

共通因数を約分する:

csc (3 θ)

= csc (3 θ) - sec (3 θ)

csc (3 θ) - sec (3 θ)

csc (3 θ) - sec (3 θ)

両辺を同じ形式にできることを証明した

\Rightarrow 真

人気の例

証明する sin(-pi/2-θ)=-cos(θ)

p ro v e sin (- \frac{π}{2} - θ) = - cos (θ)

証明する-2cos(2x)*sin(2x)=sin(-4x)

p ro v e - 2 cos (2 x) \cdot sin (2 x) = sin (- 4 x)

証明する (cos(θ^2))(1/(cos(θ^2)))=1

p ro v e (cos (θ^{2})) (\frac{1}{cos ( θ ^{2} )}) = 1

証明する sin^2(x)(1+(cos(x))/(sin(x)))=1

p ro v e sin^{2} (x) (1 + \frac{cos ( x )}{sin ( x )}) = 1

証明する tan(x)+1=1

p ro v e tan (x) + 1 = 1