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인기 있는 삼각법 >

21.16=19.6sin(x)-29.4cos(x)

해법

21.16 = 19.6 sin (x) - 29.4 cos (x)

해법

x = - 2.80086 \dots + 2 πn, x = 1.62485 \dots + 2 πn

+1

도

x = - 160.47763 \dots^{\circ} + 36 0^{\circ} n, x = 93.09749 \dots^{\circ} + 36 0^{\circ} n

솔루션 단계

21.16 = 19.6 sin (x) - 29.4 cos (x)

더하다

29.4 cos (x)

양쪽으로

19.6 sin (x) = 21.16 + 29.4 cos (x)

양쪽을 제곱

(19.6 sin (x))^{2} = (21.16 + 29.4 cos (x))^{2}

빼다

(21.16 + 29.4 cos (x))^{2}

양쪽에서

384.16 sin^{2} (x) - 447.7456 - 1244.208 cos (x) - 864.36 cos^{2} (x) = 0

삼각성을 사용하여 다시 쓰기

- 447.7456 - 1244.208 cos (x) + 384.16 sin^{2} (x) - 864.36 cos^{2} (x)

피타고라스 정체성 사용:

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

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

= - 447.7456 - 1244.208 cos (x) + 384.16 (1 - cos^{2} (x)) - 864.36 cos^{2} (x)

- 447.7456 - 1244.208 cos (x) + 384.16 (1 - cos^{2} (x)) - 864.36 cos^{2} (x)

간소화하다 :

- 1248.52 cos^{2} (x) - 1244.208 cos (x) - 63.5856

- 447.7456 - 1244.208 cos (x) + 384.16 (1 - cos^{2} (x)) - 864.36 cos^{2} (x)

384.16 (1 - cos^{2} (x))

확대한다:

384.16 - 384.16 cos^{2} (x)

384.16 (1 - cos^{2} (x))

분배 법칙 적용:

a (b - c) = ab - a c

a = 384.16, b = 1, c = cos^{2} (x)

= 384.16 \cdot 1 - 384.16 cos^{2} (x)

= 1 \cdot 384.16 - 384.16 cos^{2} (x)

숫자를 곱하시오:

1 \cdot 384.16 = 384.16

= 384.16 - 384.16 cos^{2} (x)

= - 447.7456 - 1244.208 cos (x) + 384.16 - 384.16 cos^{2} (x) - 864.36 cos^{2} (x)

- 447.7456 - 1244.208 cos (x) + 384.16 - 384.16 cos^{2} (x) - 864.36 cos^{2} (x)

단순화하세요:

- 1248.52 cos^{2} (x) - 1244.208 cos (x) - 63.5856

- 447.7456 - 1244.208 cos (x) + 384.16 - 384.16 cos^{2} (x) - 864.36 cos^{2} (x)

유사 요소 추가:

- 384.16 cos^{2} (x) - 864.36 cos^{2} (x) = - 1248.52 cos^{2} (x)

= - 447.7456 - 1244.208 cos (x) + 384.16 - 1248.52 cos^{2} (x)

집단적 용어

= - 1244.208 cos (x) - 1248.52 cos^{2} (x) - 447.7456 + 384.16

숫자 더하기/ 빼기:

- 447.7456 + 384.16 = - 63.5856

= - 1248.52 cos^{2} (x) - 1244.208 cos (x) - 63.5856

= - 1248.52 cos^{2} (x) - 1244.208 cos (x) - 63.5856

= - 1248.52 cos^{2} (x) - 1244.208 cos (x) - 63.5856

- 63.5856 - 1244.208 cos (x) - 1248.52 cos^{2} (x) = 0

대체로 해결

- 63.5856 - 1244.208 cos (x) - 1248.52 cos^{2} (x) = 0

하게:

cos (x) = u

- 63.5856 - 1244.208 u - 1248.52 u^{2} = 0

- 63.5856 - 1244.208 u - 1248.52 u^{2} = 0 : u = - \frac{1244.208 + 1230501.974016}{2497.04}, u = - \frac{1244.208 - 1230501.974016}{2497.04}

- 63.5856 - 1244.208 u - 1248.52 u^{2} = 0

표준 양식으로 작성

a x^{2} + b x + c = 0

- 1248.52 u^{2} - 1244.208 u - 63.5856 = 0

쿼드 공식으로 해결

- 1248.52 u^{2} - 1244.208 u - 63.5856 = 0

4차 방정식 공식:

위해서

a = - 1248.52, b = - 1244.208, c = - 63.5856

u_{1, 2} = \frac{- ( - 1244.208 ) \pm ( - 1244.208 ) ^{2} - 4 ( - 1248.52 ) ( - 63.5856 )}{2 ( - 1248.52 )}

u_{1, 2} = \frac{- ( - 1244.208 ) \pm ( - 1244.208 ) ^{2} - 4 ( - 1248.52 ) ( - 63.5856 )}{2 ( - 1248.52 )}

(- 1244.208)^{2} - 4 (- 1248.52) (- 63.5856) = 1230501.974016

(- 1244.208)^{2} - 4 (- 1248.52) (- 63.5856)

규칙 적용

- (- a) = a

= (- 1244.208)^{2} - 4 \cdot 1248.52 \cdot 63.5856

지수 규칙 적용:

(- a)^{n} = a^{n},

이면

n

균등하다

(- 1244.208)^{2} = 1244.20 8^{2}

= 1244.20 8^{2} - 4 \cdot 63.5856 \cdot 1248.52

숫자를 곱하시오:

4 \cdot 1248.52 \cdot 63.5856 = 317551.573248

= 1244.20 8^{2} - 317551.573248

1244.20 8^{2} = 1548053.547264

= 1548053.547264 - 317551.573248

숫자를 빼세요:

1548053.547264 - 317551.573248 = 1230501.974016

= 1230501.974016

u_{1, 2} = \frac{- ( - 1244.208 ) \pm 1230501.974016}{2 ( - 1248.52 )}

솔루션 분리

u_{1} = \frac{- ( - 1244.208 ) + 1230501.974016}{2 ( - 1248.52 )}, u_{2} = \frac{- ( - 1244.208 ) - 1230501.974016}{2 ( - 1248.52 )}

u = \frac{- ( - 1244.208 ) + 1230501.974016}{2 ( - 1248.52 )} : - \frac{1244.208 + 1230501.974016}{2497.04}

\frac{- ( - 1244.208 ) + 1230501.974016}{2 ( - 1248.52 )}

괄호 제거:

(- a) = - a, - (- a) = a

= \frac{1244.208 + 1230501.974016}{- 2 \cdot 1248.52}

숫자를 곱하시오:

2 \cdot 1248.52 = 2497.04

= \frac{1244.208 + 1230501.974016}{- 2497.04}

분수 규칙 적용:

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

= - \frac{1244.208 + 1230501.974016}{2497.04}

u = \frac{- ( - 1244.208 ) - 1230501.974016}{2 ( - 1248.52 )} : - \frac{1244.208 - 1230501.974016}{2497.04}

\frac{- ( - 1244.208 ) - 1230501.974016}{2 ( - 1248.52 )}

괄호 제거:

(- a) = - a, - (- a) = a

= \frac{1244.208 - 1230501.974016}{- 2 \cdot 1248.52}

숫자를 곱하시오:

2 \cdot 1248.52 = 2497.04

= \frac{1244.208 - 1230501.974016}{- 2497.04}

분수 규칙 적용:

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

= - \frac{1244.208 - 1230501.974016}{2497.04}

2차 방정식의 해는 다음과 같다:

u = - \frac{1244.208 + 1230501.974016}{2497.04}, u = - \frac{1244.208 - 1230501.974016}{2497.04}

뒤로 대체

u = cos (x)

cos (x) = - \frac{1244.208 + 1230501.974016}{2497.04}, cos (x) = - \frac{1244.208 - 1230501.974016}{2497.04}

cos (x) = - \frac{1244.208 + 1230501.974016}{2497.04}, cos (x) = - \frac{1244.208 - 1230501.974016}{2497.04}

cos (x) = - \frac{1244.208 + 1230501.974016}{2497.04} : x = arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn, x = - arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn

cos (x) = - \frac{1244.208 + 1230501.974016}{2497.04}

트리거 역속성 적용

cos (x) = - \frac{1244.208 + 1230501.974016}{2497.04}

일반 솔루션

cos (x) = - \frac{1244.208 + 1230501.974016}{2497.04}

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

x = arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn, x = - arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn

x = arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn, x = - arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn

cos (x) = - \frac{1244.208 - 1230501.974016}{2497.04} : x = arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn, x = - arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn

cos (x) = - \frac{1244.208 - 1230501.974016}{2497.04}

트리거 역속성 적용

cos (x) = - \frac{1244.208 - 1230501.974016}{2497.04}

일반 솔루션

cos (x) = - \frac{1244.208 - 1230501.974016}{2497.04}

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

x = arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn, x = - arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn

x = arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn, x = - arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn

모든 솔루션 결합

x = arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn, x = - arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn, x = arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn, x = - arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn

해법을 원래 방정식에 연결하여 검증

솔루션을 에 연결하여 확인합니다

19.6 sin (x) - 29.4 cos (x) = 21.16

방정식에 맞지 않는 것은 제거하십시오.

솔루션 확인

arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn :

거짓

arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn

n = 1

끼우다

arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 π 1

19.6 sin (x) - 29.4 cos (x) = 21.16

위한 {\ quad}끼우다{\ quad}

x = arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 π 1

19.6 sin (arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 π 1) - 29.4 cos (arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 π 1) = 21.16

다듬다

34.25965 \dots = 21.16

\Rightarrow 거짓

솔루션 확인

- arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn :

참

- arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn

n = 1

끼우다

- arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 π 1

19.6 sin (x) - 29.4 cos (x) = 21.16

위한 {\ quad}끼우다{\ quad}

x = - arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 π 1

19.6 sin (- arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 π 1) - 29.4 cos (- arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 π 1) = 21.16

다듬다

21.16 = 21.16

\Rightarrow 참

솔루션 확인

arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn :

참

arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn

n = 1

끼우다

arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 π 1

19.6 sin (x) - 29.4 cos (x) = 21.16

위한 {\ quad}끼우다{\ quad}

x = arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 π 1

19.6 sin (arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 π 1) - 29.4 cos (arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 π 1) = 21.16

다듬다

21.16 = 21.16

\Rightarrow 참

솔루션 확인

- arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn :

거짓

- arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn

n = 1

끼우다

- arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 π 1

19.6 sin (x) - 29.4 cos (x) = 21.16

위한 {\ quad}끼우다{\ quad}

x = - arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 π 1

19.6 sin (- arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 π 1) - 29.4 cos (- arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 π 1) = 21.16

다듬다

- 17.98273 \dots = 21.16

\Rightarrow 거짓

x = - arccos (- \frac{1244.208 + 1230501.974016}{2497.04}) + 2 πn, x = arccos (- \frac{1244.208 - 1230501.974016}{2497.04}) + 2 πn

해를 10진수 형식으로 표시

x = - 2.80086 \dots + 2 πn, x = 1.62485 \dots + 2 πn

그래프

인기 있는 예

10sin(x)cos^2(x)+3cos(x)=0

10 sin (x) cos^{2} (x) + 3 cos (x) = 0

(2 cos (x)) - 1 = 0

3cos(x)=sin^2(x)

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

sin(θ)=(5pi)/3

sin (θ) = \frac{5 π}{3}

10 cos (θ) = 8