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인기 있는 삼각법 >

(sec(x)sin(x))/(tan(x)cot(x))=sin^2(x)

해법

\frac{sec ( x ) sin ( x )}{tan ( x ) cot ( x )} = sin^{2} (x)

해법

솔루션 없음 x \in R

솔루션 단계

\frac{sec ( x ) sin ( x )}{tan ( x ) cot ( x )} = sin^{2} (x)

빼다

sin^{2} (x)

양쪽에서

\frac{sec ( x ) sin ( x )}{tan ( x ) cot ( x )} - sin^{2} (x) = 0

\frac{sec ( x ) sin ( x )}{tan ( x ) cot ( x )} - sin^{2} (x)

단순화하세요:

\frac{sec ( x ) sin ( x ) - sin ^{2} ( x ) tan ( x ) cot ( x )}{tan ( x ) cot ( x )}

\frac{sec ( x ) sin ( x )}{tan ( x ) cot ( x )} - sin^{2} (x)

요소를 분수로 변환:

sin^{2} (x) = \frac{sin ^{2} ( x ) tan ( x ) cot ( x )}{tan ( x ) cot ( x )}

= \frac{sec ( x ) sin ( x )}{tan ( x ) cot ( x )} - \frac{sin ^{2} ( x ) tan ( x ) cot ( x )}{tan ( x ) cot ( x )}

분모가 같기 때문에, 분수를 합친다:

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

= \frac{sec ( x ) sin ( x ) - sin ^{2} ( x ) tan ( x ) cot ( x )}{tan ( x ) cot ( x )}

\frac{sec ( x ) sin ( x ) - sin ^{2} ( x ) tan ( x ) cot ( x )}{tan ( x ) cot ( x )} = 0

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

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

sec (x) sin (x) - sin^{2} (x) tan (x) cot (x)

요인:

sin (x) (sec (x) - sin (x) tan (x) cot (x))

sec (x) sin (x) - sin^{2} (x) tan (x) cot (x)

지수 규칙 적용:

a^{b + c} = a^{b} a^{c}

sin^{2} (x) = sin (x) sin (x)

= sec (x) sin (x) - sin (x) sin (x) tan (x) cot (x)

공통 용어를 추출하다

sin (x)

= sin (x) (sec (x) - sin (x) tan (x) cot (x))

sin (x) (sec (x) - sin (x) tan (x) cot (x)) = 0

각 부분을 개별적으로 해결

sin (x) = 0 or sec (x) - sin (x) tan (x) cot (x) = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

일반 솔루션

sin (x) = 0

sin (x)

주기율표

2 πn

주기:

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}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn

해결 :

x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

sec (x) - sin (x) tan (x) cot (x) = 0 :

해결책 없음

sec (x) - sin (x) tan (x) cot (x) = 0

죄로 표현하라, 왜냐하면

sec (x) - cot (x) sin (x) tan (x)

기본 삼각형 항등식 사용:

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

= \frac{1}{cos ( x )} - cot (x) sin (x) tan (x)

기본 삼각형 항등식 사용:

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

= \frac{1}{cos ( x )} - \frac{cos ( x )}{sin ( x )} sin (x) tan (x)

기본 삼각형 항등식 사용:

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

= \frac{1}{cos ( x )} - \frac{cos ( x )}{sin ( x )} sin (x) \frac{sin ( x )}{cos ( x )}

\frac{1}{cos ( x )} - \frac{cos ( x )}{sin ( x )} sin (x) \frac{sin ( x )}{cos ( x )}

단순화하세요:

\frac{1 - sin ( x ) cos ( x )}{cos ( x )}

\frac{1}{cos ( x )} - \frac{cos ( x )}{sin ( x )} sin (x) \frac{sin ( x )}{cos ( x )}

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

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

다중 분수:

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

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

공통 요인 취소:

cos (x)

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

공통 요인 취소:

sin (x)

= sin (x)

= \frac{1}{cos ( x )} - sin (x)

요소를 분수로 변환:

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

= \frac{1}{cos ( x )} - \frac{sin ( x ) cos ( x )}{cos ( x )}

분모가 같기 때문에, 분수를 합친다:

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

= \frac{1 - sin ( x ) cos ( x )}{cos ( x )}

= \frac{1 - sin ( x ) cos ( x )}{cos ( x )}

\frac{1 - cos ( x ) sin ( x )}{cos ( x )} = 0

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

1 - cos (x) sin (x) = 0

삼각성을 사용하여 다시 쓰기

1 - cos (x) sin (x)

더블 앵글 아이덴티티 사용:

2 sin (x) cos (x) = sin (2 x)

sin (x) cos (x) = \frac{sin ( 2 x )}{2}

= 1 - \frac{sin ( 2 x )}{2}

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

1

를 오른쪽으로 이동

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

빼다

1

양쪽에서

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

단순화

- \frac{sin ( 2 x )}{2} = - 1

- \frac{sin ( 2 x )}{2} = - 1

양쪽을 곱한 값

2

- \frac{sin ( 2 x )}{2} = - 1

양쪽을 곱한 값

2

2 (- \frac{sin ( 2 x )}{2}) = 2 (- 1)

단순화

- sin (2 x) = - 2

- sin (2 x) = - 2

양쪽을 다음으로 나눕니다

- 1

- sin (2 x) = - 2

양쪽을 다음으로 나눕니다

- 1

\frac{- sin ( 2 x )}{- 1} = \frac{- 2}{- 1}

단순화

sin (2 x) = 2

sin (2 x) = 2

- 1 \leq sin (x) \leq 1

해결책 없음

모든 솔루션 결합

x = 2 πn, x = π + 2 πn

방정식이 정의되지 않았기 때문에:

2 πn, π + 2 πn

솔루션 없음 x \in R

그래프

인기 있는 예

sin(3x)+cos(3x)=0

sin(x)=-3/4 ,pi<x<(3pi)/4

cos(2x)+cos(x)=0,0<= x<2pi