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인기 있는 삼각법 >

tan(x/2)=sin(x)

해법

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

해법

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

+1

도

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 0^{\circ} + 72 0^{\circ} n, x = 36 0^{\circ} + 72 0^{\circ} n

솔루션 단계

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

빼다

sin (x)

양쪽에서

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

죄로 표현하라, 왜냐하면

- sin (x) + tan (\frac{x}{2})

기본 삼각형 항등식 사용:

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

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

- sin (x) + \frac{sin ( \frac{x}{2} )}{cos ( \frac{x}{2} )}

단순화하세요:

\frac{- sin ( x ) cos ( \frac{x}{2} ) + sin ( \frac{x}{2} )}{cos ( \frac{x}{2} )}

- sin (x) + \frac{sin ( \frac{x}{2} )}{cos ( \frac{x}{2} )}

요소를 분수로 변환:

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

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

분모가 같기 때문에, 분수를 합친다:

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

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

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

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

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

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

삼각성을 사용하여 다시 쓰기

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

제품을 사용하여식별요약:

sin (s) cos (t) = \frac{1}{2} (sin (s + t) + sin (s - t))

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

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

간소화하다 :

\frac{- sin ( \frac{3 x}{2} ) + sin ( \frac{x}{2} )}{2}

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

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

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

다중 분수:

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

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

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

1 \cdot (sin (x + \frac{x}{2}) + sin (x - \frac{x}{2}))

곱하다:

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

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

괄호 제거:

(a) = a

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

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

x + \frac{x}{2}

합류하다:

\frac{3 x}{2}

x + \frac{x}{2}

요소를 분수로 변환:

x = \frac{x 2}{2}

= \frac{x}{2} + \frac{x \cdot 2}{2}

분모가 같기 때문에, 분수를 합친다:

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

= \frac{x + x \cdot 2}{2}

유사 요소 추가:

x + 2 x = 3 x

= \frac{3 x}{2}

= \frac{sin ( \frac{3 x}{2} ) + sin ( x - \frac{x}{2} )}{2}

x - \frac{x}{2}

합류하다:

\frac{x}{2}

x - \frac{x}{2}

요소를 분수로 변환:

x = \frac{x 2}{2}

= - \frac{x}{2} + \frac{x \cdot 2}{2}

분모가 같기 때문에, 분수를 합친다:

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

= \frac{- x + x \cdot 2}{2}

유사 요소 추가:

- x + 2 x = x

= \frac{x}{2}

= \frac{sin ( \frac{3 x}{2} ) + sin ( \frac{x}{2} )}{2}

= sin (\frac{x}{2}) - \frac{sin ( \frac{3 x}{2} ) + sin ( \frac{x}{2} )}{2}

요소를 분수로 변환:

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

= - \frac{sin ( \frac{3 x}{2} ) + sin ( \frac{x}{2} )}{2} + \frac{sin ( \frac{x}{2} ) \cdot 2}{2}

분모가 같기 때문에, 분수를 합친다:

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

= \frac{- ( sin ( \frac{3 x}{2} ) + sin ( \frac{x}{2} ) ) + sin ( \frac{x}{2} ) \cdot 2}{2}

- (sin (\frac{3 x}{2}) + sin (\frac{x}{2})) + sin (\frac{x}{2}) \cdot 2

확대한다:

- sin (\frac{3 x}{2}) + sin (\frac{x}{2})

- (sin (\frac{3 x}{2}) + sin (\frac{x}{2})) + sin (\frac{x}{2}) \cdot 2

= - (sin (\frac{3 x}{2}) + sin (\frac{x}{2})) + 2 sin (\frac{x}{2})

- (sin (\frac{3 x}{2}) + sin (\frac{x}{2})) : - sin (\frac{3 x}{2}) - sin (\frac{x}{2})

- (sin (\frac{3 x}{2}) + sin (\frac{x}{2}))

괄호 배포

= - (sin (\frac{3 x}{2})) - (sin (\frac{x}{2}))

마이너스 플러스 규칙 적용

+ (- a) = - a

= - sin (\frac{3 x}{2}) - sin (\frac{x}{2})

= - sin (\frac{3 x}{2}) - sin (\frac{x}{2}) + sin (\frac{x}{2}) \cdot 2

유사 요소 추가:

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

= - sin (\frac{3 x}{2}) + sin (\frac{x}{2})

= \frac{- sin ( \frac{3 x}{2} ) + sin ( \frac{x}{2} )}{2}

= \frac{- sin ( \frac{3 x}{2} ) + sin ( \frac{x}{2} )}{2}

제품식별에 대한 합계 사용:

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

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

\frac{2 sin ( \frac{\frac{x}{2} - \frac{3 x}{2}}{2} ) cos ( \frac{\frac{x}{2} + \frac{3 x}{2}}{2} )}{2}

간소화하다 :

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

\frac{2 sin ( \frac{\frac{x}{2} - \frac{3 x}{2}}{2} ) cos ( \frac{\frac{x}{2} + \frac{3 x}{2}}{2} )}{2}

분수를 합치다

\frac{x}{2} + \frac{3 x}{2} : 2 x

규칙 적용

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

= \frac{x + 3 x}{2}

유사 요소 추가:

x + 3 x = 4 x

= \frac{4 x}{2}

숫자를 나눕니다:

\frac{4}{2} = 2

= 2 x

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

분수를 합치다

\frac{x}{2} - \frac{3 x}{2} : - x

규칙 적용

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

= \frac{x - 3 x}{2}

유사 요소 추가:

x - 3 x = - 2 x

= \frac{- 2 x}{2}

분수 규칙 적용:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{2 x}{2}

숫자를 나눕니다:

\frac{2}{2} = 1

= - x

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

분수 규칙 적용:

\frac{- a}{b} = - \frac{a}{b}

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

숫자를 나눕니다:

\frac{2}{2} = 1

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

네거티브 각도식별사용:

sin (- x) = - sin (x)

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

괄호 제거:

(- a) = - a

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

숫자를 나눕니다:

\frac{2}{2} = 1

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

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

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

각 부분을 개별적으로 해결

cos (x) = 0 or sin (\frac{x}{2}) = 0

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

일반 솔루션

cos (x) = 0

cos (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} 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}

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

sin (\frac{x}{2}) = 0 : x = 4 πn, x = 2 π + 4 πn

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

일반 솔루션

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

\frac{x}{2} = 0 + 2 πn, \frac{x}{2} = π + 2 πn

\frac{x}{2} = 0 + 2 πn, \frac{x}{2} = π + 2 πn

\frac{x}{2} = 0 + 2 πn

해결 :

x = 4 πn

\frac{x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{x}{2} = 2 πn

양쪽을 곱한 값

2

\frac{x}{2} = 2 πn

양쪽을 곱한 값

2

\frac{2 x}{2} = 2 \cdot 2 πn

단순화

x = 4 πn

x = 4 πn

\frac{x}{2} = π + 2 πn

해결 :

x = 2 π + 4 πn

\frac{x}{2} = π + 2 πn

양쪽을 곱한 값

2

\frac{x}{2} = π + 2 πn

양쪽을 곱한 값

2

\frac{2 x}{2} = 2 π + 2 \cdot 2 πn

단순화

x = 2 π + 4 πn

x = 2 π + 4 πn

x = 4 πn, x = 2 π + 4 πn

모든 솔루션 결합

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

그래프

인기 있는 예

7sin(B)+sqrt(23)=0

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7sin(θ)-1=5sin(θ)

7 sin (θ) - 1 = 5 sin (θ)

csc (θ) = - \frac{7}{3}