What is the general solution for 7sin(x)=cos(x)-4 ?

The general solution for 7sin(x)=cos(x)-4 is x=-2.39843…+2pin,x=2pi-0.45936…+2pin

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7sin(x)=cos(x)-4

Solution

7 sin (x) = cos (x) - 4

Solution

x = - 2.39843 \dots + 2 πn, x = 2 π - 0.45936 \dots + 2 πn

Degrees

x = - 137.41999 \dots^{\circ} + 36 0^{\circ} n, x = 333.68020 \dots^{\circ} + 36 0^{\circ} n

Solution steps

7 sin (x) = cos (x) - 4

Square both sides

(7 sin (x))^{2} = (cos (x) - 4)^{2}

Subtract

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

from both sides

49 sin^{2} (x) - cos^{2} (x) + 8 cos (x) - 16 = 0

Rewrite using trig identities

- 16 - cos^{2} (x) + 49 sin^{2} (x) + 8 cos (x)

Use the Pythagorean identity:

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

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

= - 16 - cos^{2} (x) + 49 (1 - cos^{2} (x)) + 8 cos (x)

Simplify

- 16 - cos^{2} (x) + 49 (1 - cos^{2} (x)) + 8 cos (x) : 8 cos (x) - 50 cos^{2} (x) + 33

- 16 - cos^{2} (x) + 49 (1 - cos^{2} (x)) + 8 cos (x)

Expand

49 (1 - cos^{2} (x)) : 49 - 49 cos^{2} (x)

49 (1 - cos^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 49, b = 1, c = cos^{2} (x)

= 49 \cdot 1 - 49 cos^{2} (x)

Multiply the numbers:

49 \cdot 1 = 49

= 49 - 49 cos^{2} (x)

= - 16 - cos^{2} (x) + 49 - 49 cos^{2} (x) + 8 cos (x)

Simplify

- 16 - cos^{2} (x) + 49 - 49 cos^{2} (x) + 8 cos (x) : 8 cos (x) - 50 cos^{2} (x) + 33

- 16 - cos^{2} (x) + 49 - 49 cos^{2} (x) + 8 cos (x)

Group like terms

= - cos^{2} (x) - 49 cos^{2} (x) + 8 cos (x) - 16 + 49

Add similar elements:

- cos^{2} (x) - 49 cos^{2} (x) = - 50 cos^{2} (x)

= - 50 cos^{2} (x) + 8 cos (x) - 16 + 49

Add/Subtract the numbers:

- 16 + 49 = 33

= 8 cos (x) - 50 cos^{2} (x) + 33

= 8 cos (x) - 50 cos^{2} (x) + 33

= 8 cos (x) - 50 cos^{2} (x) + 33

33 - 50 cos^{2} (x) + 8 cos (x) = 0

Solve by substitution

33 - 50 cos^{2} (x) + 8 cos (x) = 0

Let:

cos (x) = u

33 - 50 u^{2} + 8 u = 0

33 - 50 u^{2} + 8 u = 0 : u = - \frac{- 4 + 7 34}{50}, u = \frac{4 + 7 34}{50}

33 - 50 u^{2} + 8 u = 0

Write in the standard form

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

- 50 u^{2} + 8 u + 33 = 0

Solve with the quadratic formula

- 50 u^{2} + 8 u + 33 = 0

Quadratic Equation Formula:

For

a = - 50, b = 8, c = 33

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 ( - 50 ) \cdot 33}{2 ( - 50 )}

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 ( - 50 ) \cdot 33}{2 ( - 50 )}

8^{2} - 4 (- 50) \cdot 33 = 1434

8^{2} - 4 (- 50) \cdot 33

Apply rule

- (- a) = a

= 8^{2} + 4 \cdot 50 \cdot 33

Multiply the numbers:

4 \cdot 50 \cdot 33 = 6600

= 8^{2} + 6600

8^{2} = 64

= 64 + 6600

Add the numbers:

64 + 6600 = 6664

= 6664

Prime factorization of

6664 : 2^{3} \cdot 7^{2} \cdot 17

6664

6664

divides by

26664 = 3332 \cdot 2

= 2 \cdot 3332

3332

divides by

23332 = 1666 \cdot 2

= 2 \cdot 2 \cdot 1666

1666

divides by

21666 = 833 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 833

833

divides by

7833 = 119 \cdot 7

= 2 \cdot 2 \cdot 2 \cdot 7 \cdot 119

119

divides by

7119 = 17 \cdot 7

= 2 \cdot 2 \cdot 2 \cdot 7 \cdot 7 \cdot 17

2, 7, 17

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 7 \cdot 7 \cdot 17

= 2^{3} \cdot 7^{2} \cdot 17

= 2^{3} \cdot 7^{2} \cdot 17

Apply exponent rule:

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

= 2^{2} \cdot 7^{2} \cdot 2 \cdot 17

Apply radical rule:

= 2^{2} 7^{2} 2 \cdot 17

Apply radical rule:

2^{2} = 2

= 2 7^{2} 2 \cdot 17

Apply radical rule:

7^{2} = 7

= 2 \cdot 7 2 \cdot 17

Refine

= 1434

u_{1, 2} = \frac{- 8 \pm 14 34}{2 ( - 50 )}

Separate the solutions

u_{1} = \frac{- 8 + 14 34}{2 ( - 50 )}, u_{2} = \frac{- 8 - 14 34}{2 ( - 50 )}

u = \frac{- 8 + 14 34}{2 ( - 50 )} : - \frac{- 4 + 7 34}{50}

\frac{- 8 + 14 34}{2 ( - 50 )}

Remove parentheses:

(- a) = - a

= \frac{- 8 + 14 34}{- 2 \cdot 50}

Multiply the numbers:

2 \cdot 50 = 100

= \frac{- 8 + 14 34}{- 100}

Apply the fraction rule:

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

= - \frac{- 8 + 14 34}{100}

Cancel

\frac{- 8 + 14 34}{100} : \frac{7 34 - 4}{50}

\frac{- 8 + 14 34}{100}

Factor

- 8 + 1434 : 2 (- 4 + 734)

- 8 + 1434

Rewrite as

= - 2 \cdot 4 + 2 \cdot 734

Factor out common term

2

= 2 (- 4 + 734)

= \frac{2 ( - 4 + 7 34 )}{100}

Cancel the common factor:

2

= \frac{- 4 + 7 34}{50}

= - \frac{7 34 - 4}{50}

= - \frac{- 4 + 7 34}{50}

u = \frac{- 8 - 14 34}{2 ( - 50 )} : \frac{4 + 7 34}{50}

\frac{- 8 - 14 34}{2 ( - 50 )}

Remove parentheses:

(- a) = - a

= \frac{- 8 - 14 34}{- 2 \cdot 50}

Multiply the numbers:

2 \cdot 50 = 100

= \frac{- 8 - 14 34}{- 100}

Apply the fraction rule:

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

- 8 - 1434 = - (8 + 1434)

= \frac{8 + 14 34}{100}

Factor

8 + 1434 : 2 (4 + 734)

8 + 1434

Rewrite as

= 2 \cdot 4 + 2 \cdot 734

Factor out common term

2

= 2 (4 + 734)

= \frac{2 ( 4 + 7 34 )}{100}

Cancel the common factor:

2

= \frac{4 + 7 34}{50}

The solutions to the quadratic equation are:

u = - \frac{- 4 + 7 34}{50}, u = \frac{4 + 7 34}{50}

Substitute back

u = cos (x)

cos (x) = - \frac{- 4 + 7 34}{50}, cos (x) = \frac{4 + 7 34}{50}

cos (x) = - \frac{- 4 + 7 34}{50}, cos (x) = \frac{4 + 7 34}{50}

cos (x) = - \frac{- 4 + 7 34}{50} : x = arccos (- \frac{- 4 + 7 34}{50}) + 2 πn, x = - arccos (- \frac{- 4 + 7 34}{50}) + 2 πn

cos (x) = - \frac{- 4 + 7 34}{50}

Apply trig inverse properties

cos (x) = - \frac{- 4 + 7 34}{50}

General solutions for

cos (x) = - \frac{- 4 + 7 34}{50}

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

x = arccos (- \frac{- 4 + 7 34}{50}) + 2 πn, x = - arccos (- \frac{- 4 + 7 34}{50}) + 2 πn

x = arccos (- \frac{- 4 + 7 34}{50}) + 2 πn, x = - arccos (- \frac{- 4 + 7 34}{50}) + 2 πn

cos (x) = \frac{4 + 7 34}{50} : x = arccos (\frac{4 + 7 34}{50}) + 2 πn, x = 2 π - arccos (\frac{4 + 7 34}{50}) + 2 πn

cos (x) = \frac{4 + 7 34}{50}

Apply trig inverse properties

cos (x) = \frac{4 + 7 34}{50}

General solutions for

cos (x) = \frac{4 + 7 34}{50}

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

x = arccos (\frac{4 + 7 34}{50}) + 2 πn, x = 2 π - arccos (\frac{4 + 7 34}{50}) + 2 πn

x = arccos (\frac{4 + 7 34}{50}) + 2 πn, x = 2 π - arccos (\frac{4 + 7 34}{50}) + 2 πn

Combine all the solutions

x = arccos (- \frac{- 4 + 7 34}{50}) + 2 πn, x = - arccos (- \frac{- 4 + 7 34}{50}) + 2 πn, x = arccos (\frac{4 + 7 34}{50}) + 2 πn, x = 2 π - arccos (\frac{4 + 7 34}{50}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

7 sin (x) = cos (x) - 4

Remove the ones that don't agree with the equation.

Check the solution

arccos (- \frac{- 4 + 7 34}{50}) + 2 πn :

False

arccos (- \frac{- 4 + 7 34}{50}) + 2 πn

Plug in

n = 1

arccos (- \frac{- 4 + 7 34}{50}) + 2 π 1

For

7 sin (x) = cos (x) - 4

plug in

x = arccos (- \frac{- 4 + 7 34}{50}) + 2 π 1

7 sin (arccos (- \frac{- 4 + 7 34}{50}) + 2 π 1) = cos (arccos (- \frac{- 4 + 7 34}{50}) + 2 π 1) - 4

Refine

4.73633 \dots = - 4.73633 \dots

\Rightarrow False

Check the solution

- arccos (- \frac{- 4 + 7 34}{50}) + 2 πn :

True

- arccos (- \frac{- 4 + 7 34}{50}) + 2 πn

Plug in

n = 1

- arccos (- \frac{- 4 + 7 34}{50}) + 2 π 1

For

7 sin (x) = cos (x) - 4

plug in

x = - arccos (- \frac{- 4 + 7 34}{50}) + 2 π 1

7 sin (- arccos (- \frac{- 4 + 7 34}{50}) + 2 π 1) = cos (- arccos (- \frac{- 4 + 7 34}{50}) + 2 π 1) - 4

Refine

- 4.73633 \dots = - 4.73633 \dots

\Rightarrow True

Check the solution

arccos (\frac{4 + 7 34}{50}) + 2 πn :

False

arccos (\frac{4 + 7 34}{50}) + 2 πn

Plug in

n = 1

arccos (\frac{4 + 7 34}{50}) + 2 π 1

For

7 sin (x) = cos (x) - 4

plug in

x = arccos (\frac{4 + 7 34}{50}) + 2 π 1

7 sin (arccos (\frac{4 + 7 34}{50}) + 2 π 1) = cos (arccos (\frac{4 + 7 34}{50}) + 2 π 1) - 4

Refine

3.10366 \dots = - 3.10366 \dots

\Rightarrow False

Check the solution

2 π - arccos (\frac{4 + 7 34}{50}) + 2 πn :

True

2 π - arccos (\frac{4 + 7 34}{50}) + 2 πn

Plug in

n = 1

2 π - arccos (\frac{4 + 7 34}{50}) + 2 π 1

For

7 sin (x) = cos (x) - 4

plug in

x = 2 π - arccos (\frac{4 + 7 34}{50}) + 2 π 1

7 sin (2 π - arccos (\frac{4 + 7 34}{50}) + 2 π 1) = cos (2 π - arccos (\frac{4 + 7 34}{50}) + 2 π 1) - 4

Refine

- 3.10366 \dots = - 3.10366 \dots

\Rightarrow True

x = - arccos (- \frac{- 4 + 7 34}{50}) + 2 πn, x = 2 π - arccos (\frac{4 + 7 34}{50}) + 2 πn

Show solutions in decimal form

x = - 2.39843 \dots + 2 πn, x = 2 π - 0.45936 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 7sin(x)=cos(x)-4 ?
The general solution for 7sin(x)=cos(x)-4 is x=-2.39843…+2pin,x=2pi-0.45936…+2pin