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인기 있는 삼각법 >

증명 sin^6(a)+cos^6(a)=1-3sin^2(a)*cos^2(a)

해법

증명

sin^{6} (a) + cos^{6} (a) = 1 - 3 sin^{2} (a) \cdot cos^{2} (a)

해법

참

솔루션 단계

sin^{6} (a) + cos^{6} (a) = 1 - 3 sin^{2} (a) cos^{2} (a)

왼쪽 조작

sin^{6} (a) + cos^{6} (a)

cos^{6} (a) + sin^{6} (a)

요인:

(cos^{2} (a) + sin^{2} (a)) (cos^{4} (a) - sin^{2} (a) cos^{2} (a) + sin^{4} (a))

cos^{6} (a) + sin^{6} (a)

cos^{6} (a) + sin^{6} (a) (cos^{2} (a))^{3} + (sin^{2} (a))^{3}

로 다시 씁니다

cos^{6} (a) + sin^{6} (a)

지수 규칙 적용:

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

sin^{6} (a) = (sin^{2} (a))^{3}

= cos^{6} (a) + (sin^{2} (a))^{3}

지수 규칙 적용:

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

cos^{6} (a) = (cos^{2} (a))^{3}

= (cos^{2} (a))^{3} + (sin^{2} (a))^{3}

= (cos^{2} (a))^{3} + (sin^{2} (a))^{3}

입방체의 합 공식 적용:

x^{3} + y^{3} = (x + y) (x^{2} - x y + y^{2})

(cos^{2} (a))^{3} + (sin^{2} (a))^{3} = (cos^{2} (a) + sin^{2} (a)) (cos^{4} (a) - sin^{2} (a) cos^{2} (a) + sin^{4} (a))

= (sin^{2} (a) + cos^{2} (a)) (sin^{4} (a) - sin^{2} (a) cos^{2} (a) + cos^{4} (a))

= (cos^{2} (a) + sin^{2} (a)) (cos^{4} (a) + sin^{4} (a) - cos^{2} (a) sin^{2} (a))

삼각성을 사용하여 다시 쓰기

(cos^{2} (a) + sin^{2} (a)) (cos^{4} (a) + sin^{4} (a) - cos^{2} (a) sin^{2} (a))

피타고라스 정체성 사용:

cos^{2} (x) + sin^{2} (x) = 1

= 1 \cdot (cos^{4} (a) + sin^{4} (a) - cos^{2} (a) sin^{2} (a))

단순화

= cos^{4} (a) + sin^{4} (a) - cos^{2} (a) sin^{2} (a)

더블 앵글 아이덴티티 사용:

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

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

= cos^{4} (a) + sin^{4} (a) + cos^{2} (a) (cos (2 a) - cos^{2} (a))

cos^{4} (a) + sin^{4} (a) + cos^{2} (a) (cos (2 a) - cos^{2} (a))

단순화하세요:

sin^{4} (a) + cos^{2} (a) cos (2 a)

cos^{4} (a) + sin^{4} (a) + cos^{2} (a) (cos (2 a) - cos^{2} (a))

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

확대한다:

cos^{2} (a) cos (2 a) - cos^{4} (a)

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

분배 법칙 적용:

a (b - c) = ab - a c

a = cos^{2} (a), b = cos (2 a), c = cos^{2} (a)

= cos^{2} (a) cos (2 a) - cos^{2} (a) cos^{2} (a)

cos^{2} (a) cos^{2} (a) = cos^{4} (a)

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

지수 규칙 적용:

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

cos^{2} (a) cos^{2} (a) = cos^{2 + 2} (a)

= cos^{2 + 2} (a)

숫자 추가:

2 + 2 = 4

= cos^{4} (a)

= cos^{2} (a) cos (2 a) - cos^{4} (a)

= cos^{4} (a) + sin^{4} (a) + cos^{2} (a) cos (2 a) - cos^{4} (a)

유사 요소 추가:

cos^{4} (a) - cos^{4} (a) = 0

= sin^{4} (a) + cos^{2} (a) cos (2 a)

= sin^{4} (a) + cos^{2} (a) cos (2 a)

더블 앵글 아이덴티티 사용:

cos (2 x) = 1 - 2 sin^{2} (x)

= sin^{4} (a) + cos^{2} (a) (1 - 2 sin^{2} (a))

피타고라스 정체성 사용:

cos^{2} (x) + sin^{2} (x) = 1

cos^{2} (x) = 1 - sin^{2} (x)

= sin^{4} (a) + (1 - sin^{2} (a)) (1 - 2 sin^{2} (a))

sin^{4} (a) + (1 - sin^{2} (a)) (1 - 2 sin^{2} (a))

단순화하세요:

3 sin^{4} (a) - 3 sin^{2} (a) + 1

sin^{4} (a) + (1 - sin^{2} (a)) (1 - 2 sin^{2} (a))

(1 - sin^{2} (a)) (1 - 2 sin^{2} (a))

확대한다:

1 - 3 sin^{2} (a) + 2 sin^{4} (a)

(1 - sin^{2} (a)) (1 - 2 sin^{2} (a))

호일 방법 적용:

(a + b) (c + d) = a c + a d + b c + b d

a = 1, b = - sin^{2} (a), c = 1, d = - 2 sin^{2} (a)

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

마이너스 플러스 규칙 적용

+ (- a) = - a, (- a) (- b) = ab

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

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

단순화하세요:

1 - 3 sin^{2} (a) + 2 sin^{4} (a)

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

1 \cdot 1 = 1

1 \cdot 1

숫자를 곱하시오:

1 \cdot 1 = 1

= 1

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

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

숫자를 곱하시오:

1 \cdot 2 = 2

= 2 sin^{2} (a)

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

1 \cdot sin^{2} (a)

곱하다:

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

= sin^{2} (a)

2 sin^{2} (a) sin^{2} (a) = 2 sin^{4} (a)

2 sin^{2} (a) sin^{2} (a)

지수 규칙 적용:

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

sin^{2} (a) sin^{2} (a) = sin^{2 + 2} (a)

= 2 sin^{2 + 2} (a)

숫자 추가:

2 + 2 = 4

= 2 sin^{4} (a)

= 1 - 2 sin^{2} (a) - sin^{2} (a) + 2 sin^{4} (a)

유사 요소 추가:

- 2 sin^{2} (a) - sin^{2} (a) = - 3 sin^{2} (a)

= 1 - 3 sin^{2} (a) + 2 sin^{4} (a)

= 1 - 3 sin^{2} (a) + 2 sin^{4} (a)

= sin^{4} (a) + 1 - 3 sin^{2} (a) + 2 sin^{4} (a)

sin^{4} (a) + 1 - 3 sin^{2} (a) + 2 sin^{4} (a)

단순화하세요:

3 sin^{4} (a) - 3 sin^{2} (a) + 1

sin^{4} (a) + 1 - 3 sin^{2} (a) + 2 sin^{4} (a)

집단적 용어

= sin^{4} (a) - 3 sin^{2} (a) + 2 sin^{4} (a) + 1

유사 요소 추가:

sin^{4} (a) + 2 sin^{4} (a) = 3 sin^{4} (a)

= 3 sin^{4} (a) - 3 sin^{2} (a) + 1

= 3 sin^{4} (a) - 3 sin^{2} (a) + 1

= 3 sin^{4} (a) - 3 sin^{2} (a) + 1

= 3 sin^{4} (a) - 3 sin^{2} (a) + 1

- 3 sin^{2} (a) + 3 sin^{4} (a)

요인:

3 sin^{2} (a) (sin (a) + 1) (sin (a) - 1)

- 3 sin^{2} (a) + 3 sin^{4} (a)

공통 용어를 추출하다

3 sin^{2} (a) : 3 sin^{2} (a) (sin^{2} (a) - 1)

3 sin^{4} (a) - 3 sin^{2} (a)

지수 규칙 적용:

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

sin^{4} (a) = sin^{2} (a) sin^{2} (a)

= 3 sin^{2} (a) sin^{2} (a) - 3 sin^{2} (a)

공통 용어를 추출하다

3 sin^{2} (a)

= 3 sin^{2} (a) (sin^{2} (a) - 1)

= 3 sin^{2} (a) (sin^{2} (a) - 1)

sin^{2} (a) - 1

요인:

(sin (a) + 1) (sin (a) - 1)

sin^{2} (a) - 1

1 1^{2}

로 다시 씁니다

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

두 제곱 공식의 차이 적용:

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

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

= (sin (a) + 1) (sin (a) - 1)

= 3 sin^{2} (a) (sin (a) + 1) (sin (a) - 1)

= 1 + (- 1 + sin (a)) (1 + sin (a)) \cdot 3 sin^{2} (a)

삼각성을 사용하여 다시 쓰기

1 + (- 1 + sin (a)) (1 + sin (a)) \cdot 3 sin^{2} (a)

(sin (a) + 1) (sin (a) - 1)

확대한다:

sin^{2} (a) - 1

(sin (a) + 1) (sin (a) - 1)

두 제곱 공식의 차이 적용:

(a + b) (a - b) = a^{2} - b^{2}

a = sin (a), b = 1

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

규칙 적용

1^{a} = 1

1^{2} = 1

= sin^{2} (a) - 1

= 1 + 3 sin^{2} (a) (sin^{2} (a) - 1)

피타고라스 정체성 사용:

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

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

= 1 + 3 sin^{2} (a) (- cos^{2} (a))

단순화

= 1 - 3 sin^{2} (a) cos^{2} (a)

= 1 - 3 sin^{2} (a) cos^{2} (a)

우리는 양측이 같은 형태를 취할 수 있다는 것을 보여주었습니다

\Rightarrow 참

인기 있는 예

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