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人気のある三角関数 >

sin^2(a)sec(a)=tan^2(a)

解

sin^{2} (a) sec (a) = tan^{2} (a)

解

a = 2 πn, a = π + 2 πn

+1

度

a = 0^{\circ} + 36 0^{\circ} n, a = 18 0^{\circ} + 36 0^{\circ} n

解答ステップ

sin^{2} (a) sec (a) = tan^{2} (a)

両辺から

tan^{2} (a)

を引く

sin^{2} (a) sec (a) - tan^{2} (a) = 0

サイン, コサインで表わす

- tan^{2} (a) + sec (a) sin^{2} (a)

基本的な三角関数の公式を使用する:

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

= - (\frac{sin ( a )}{cos ( a )})^{2} + sec (a) sin^{2} (a)

基本的な三角関数の公式を使用する:

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

= - (\frac{sin ( a )}{cos ( a )})^{2} + \frac{1}{cos ( a )} sin^{2} (a)

簡素化

- (\frac{sin ( a )}{cos ( a )})^{2} + \frac{1}{cos ( a )} sin^{2} (a) : \frac{- sin ^{2} ( a ) + sin ^{2} ( a ) cos ( a )}{cos ^{2} ( a )}

- (\frac{sin ( a )}{cos ( a )})^{2} + \frac{1}{cos ( a )} sin^{2} (a)

(\frac{sin ( a )}{cos ( a )})^{2} = \frac{sin ^{2} ( a )}{cos ^{2} ( a )}

(\frac{sin ( a )}{cos ( a )})^{2}

指数の規則を適用する:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{sin ^{2} ( a )}{cos ^{2} ( a )}

\frac{1}{cos ( a )} sin^{2} (a) = \frac{sin ^{2} ( a )}{cos ( a )}

\frac{1}{cos ( a )} sin^{2} (a)

分数を乗じる:

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

= \frac{1 \cdot sin ^{2} ( a )}{cos ( a )}

乗算:

1 \cdot sin^{2} (a) = sin^{2} (a)

= \frac{sin ^{2} ( a )}{cos ( a )}

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

以下の最小公倍数:

cos^{2} (a), cos (a) : cos^{2} (a)

cos^{2} (a), cos (a)

最小公倍数 (LCM)

cos^{2} (a)

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 (a)

= cos^{2} (a)

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

cos^{2} (a)

\frac{sin ^{2} ( a )}{cos ( a )}

の場合

:

分母と分子に以下を乗じる:

cos (a)

\frac{sin ^{2} ( a )}{cos ( a )} = \frac{sin ^{2} ( a ) cos ( a )}{cos ( a ) cos ( a )} = \frac{sin ^{2} ( a ) cos ( a )}{cos ^{2} ( a )}

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

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{- sin ^{2} ( a ) + sin ^{2} ( a ) cos ( a )}{cos ^{2} ( a )}

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

\frac{- sin ^{2} ( a ) + cos ( a ) sin ^{2} ( a )}{cos ^{2} ( a )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

- sin^{2} (a) + cos (a) sin^{2} (a) = 0

因数

- sin^{2} (a) + cos (a) sin^{2} (a) : sin^{2} (a) (cos (a) - 1)

- sin^{2} (a) + cos (a) sin^{2} (a)

共通項をくくり出す

sin^{2} (a)

= sin^{2} (a) (- 1 + cos (a))

sin^{2} (a) (cos (a) - 1) = 0

各部分を別個に解く

sin^{2} (a) = 0 or cos (a) - 1 = 0

sin^{2} (a) = 0 : a = 2 πn, a = π + 2 πn

sin^{2} (a) = 0

規則を適用

x^{n} = 0 \Rightarrow x = 0

sin (a) = 0

以下の一般解

sin (a) = 0

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

a = 0 + 2 πn, a = π + 2 πn

a = 0 + 2 πn, a = π + 2 πn

解く

a = 0 + 2 πn : a = 2 πn

a = 0 + 2 πn

0 + 2 πn = 2 πn

a = 2 πn

a = 2 πn, a = π + 2 πn

cos (a) - 1 = 0 : a = 2 πn

cos (a) - 1 = 0

1

を右側に移動します

cos (a) - 1 = 0

両辺に

1

を足す

cos (a) - 1 + 1 = 0 + 1

簡素化

cos (a) = 1

cos (a) = 1

以下の一般解

cos (a) = 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}

a = 0 + 2 πn

a = 0 + 2 πn

解く

a = 0 + 2 πn : a = 2 πn

a = 0 + 2 πn

0 + 2 πn = 2 πn

a = 2 πn

a = 2 πn

すべての解を組み合わせる

a = 2 πn, a = π + 2 πn

グラフ

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