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인기 있는 삼각법 >

(cos^2(x)+1)/(1+cot^2(x))=1

해법

\frac{cos ^{2} ( x ) + 1}{1 + cot ^{2} ( x )} = 1

해법

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

+1

도

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

솔루션 단계

\frac{cos ^{2} ( x ) + 1}{1 + cot ^{2} ( x )} = 1

빼다

1

양쪽에서

\frac{cos ^{2} ( x ) + 1}{1 + cot ^{2} ( x )} - 1 = 0

\frac{cos ^{2} ( x ) + 1}{1 + cot ^{2} ( x )} - 1

단순화하세요:

\frac{cos ^{2} ( x ) - cot ^{2} ( x )}{1 + cot ^{2} ( x )}

\frac{cos ^{2} ( x ) + 1}{1 + cot ^{2} ( x )} - 1

요소를 분수로 변환:

1 = \frac{1 ( 1 + cot ^{2} ( x ) )}{1 + cot ^{2} ( x )}

= \frac{cos ^{2} ( x ) + 1}{1 + cot ^{2} ( x )} - \frac{1 \cdot ( 1 + cot ^{2} ( x ) )}{1 + cot ^{2} ( x )}

분모가 같기 때문에, 분수를 합친다:

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

= \frac{cos ^{2} ( x ) + 1 - 1 \cdot ( 1 + cot ^{2} ( x ) )}{1 + cot ^{2} ( x )}

곱하다:

1 \cdot (1 + cot^{2} (x)) = (1 + cot^{2} (x))

= \frac{cos ^{2} ( x ) + 1 - ( cot ^{2} ( x ) + 1 )}{1 + cot ^{2} ( x )}

cos^{2} (x) + 1 - (1 + cot^{2} (x))

확대한다:

cos^{2} (x) - cot^{2} (x)

cos^{2} (x) + 1 - (1 + cot^{2} (x))

- (1 + cot^{2} (x)) : - 1 - cot^{2} (x)

- (1 + cot^{2} (x))

괄호 배포

= - (1) - (cot^{2} (x))

마이너스 플러스 규칙 적용

+ (- a) = - a

= - 1 - cot^{2} (x)

= cos^{2} (x) + 1 - 1 - cot^{2} (x)

1 - 1 = 0

= cos^{2} (x) - cot^{2} (x)

= \frac{cos ^{2} ( x ) - cot ^{2} ( x )}{1 + cot ^{2} ( x )}

\frac{cos ^{2} ( x ) - cot ^{2} ( x )}{1 + cot ^{2} ( x )} = 0

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

cos^{2} (x) - cot^{2} (x) = 0

cos^{2} (x) - cot^{2} (x)

요인:

(cos (x) + cot (x)) (cos (x) - cot (x))

cos^{2} (x) - cot^{2} (x)

두 제곱 공식의 차이 적용:

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

cos^{2} (x) - cot^{2} (x) = (cos (x) + cot (x)) (cos (x) - cot (x))

= (cos (x) + cot (x)) (cos (x) - cot (x))

(cos (x) + cot (x)) (cos (x) - cot (x)) = 0

각 부분을 개별적으로 해결

cos (x) + cot (x) = 0 or cos (x) - cot (x) = 0

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

cos (x) + cot (x) = 0

죄로 표현하라, 왜냐하면

cos (x) + cot (x)

기본 삼각형 항등식 사용:

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

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

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

단순화하세요:

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

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

요소를 분수로 변환:

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

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

분모가 같기 때문에, 분수를 합친다:

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

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

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

\frac{cos ( x ) + cos ( x ) sin ( x )}{sin ( x )} = 0

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

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

cos (x) + cos (x) sin (x)

요인:

cos (x) (sin (x) + 1)

cos (x) + cos (x) sin (x)

공통 용어를 추출하다

cos (x)

= cos (x) (1 + sin (x))

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

각 부분을 개별적으로 해결

cos (x) = 0 or sin (x) + 1 = 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 (x) + 1 = 0 : x = \frac{3 π}{2} + 2 πn

sin (x) + 1 = 0

1

를 오른쪽으로 이동

sin (x) + 1 = 0

빼다

1

양쪽에서

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

단순화

sin (x) = - 1

sin (x) = - 1

일반 솔루션

sin (x) = - 1

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 = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

모든 솔루션 결합

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

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

cos (x) - cot (x) = 0

죄로 표현하라, 왜냐하면

cos (x) - cot (x)

기본 삼각형 항등식 사용:

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

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

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

단순화하세요:

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

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

요소를 분수로 변환:

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

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

분모가 같기 때문에, 분수를 합친다:

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

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

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

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

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

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

- cos (x) + cos (x) sin (x)

요인:

cos (x) (sin (x) - 1)

- cos (x) + cos (x) sin (x)

공통 용어를 추출하다

cos (x)

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

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

각 부분을 개별적으로 해결

cos (x) = 0 or sin (x) - 1 = 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 (x) - 1 = 0 : x = \frac{π}{2} + 2 πn

sin (x) - 1 = 0

1

를 오른쪽으로 이동

sin (x) - 1 = 0

더하다

1

양쪽으로

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

단순화

sin (x) = 1

sin (x) = 1

일반 솔루션

sin (x) = 1

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 = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

모든 솔루션 결합

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

모든 솔루션 결합

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

그래프

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