What is the general solution for 9sin(x)=cos(x)-7 ?

The general solution for 9sin(x)=cos(x)-7 is x=-2.14734…+2pin,x=2pi-0.77293…+2pin

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Popular Trigonometry >

9sin(x)=cos(x)-7

Solution

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

Solution

x = - 2.14734 \dots + 2 πn, x = 2 π - 0.77293 \dots + 2 πn

Degrees

x = - 123.03388 \dots^{\circ} + 36 0^{\circ} n, x = 315.71426 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Square both sides

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

Subtract

(cos (x) - 7)^{2}

from both sides

81 sin^{2} (x) - cos^{2} (x) + 14 cos (x) - 49 = 0

Rewrite using trig identities

- 49 - cos^{2} (x) + 14 cos (x) + 81 sin^{2} (x)

Use the Pythagorean identity:

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

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

= - 49 - cos^{2} (x) + 14 cos (x) + 81 (1 - cos^{2} (x))

Simplify

- 49 - cos^{2} (x) + 14 cos (x) + 81 (1 - cos^{2} (x)) : 14 cos (x) - 82 cos^{2} (x) + 32

- 49 - cos^{2} (x) + 14 cos (x) + 81 (1 - cos^{2} (x))

Expand

81 (1 - cos^{2} (x)) : 81 - 81 cos^{2} (x)

81 (1 - cos^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 81, b = 1, c = cos^{2} (x)

= 81 \cdot 1 - 81 cos^{2} (x)

Multiply the numbers:

81 \cdot 1 = 81

= 81 - 81 cos^{2} (x)

= - 49 - cos^{2} (x) + 14 cos (x) + 81 - 81 cos^{2} (x)

Simplify

- 49 - cos^{2} (x) + 14 cos (x) + 81 - 81 cos^{2} (x) : 14 cos (x) - 82 cos^{2} (x) + 32

- 49 - cos^{2} (x) + 14 cos (x) + 81 - 81 cos^{2} (x)

Group like terms

= - cos^{2} (x) + 14 cos (x) - 81 cos^{2} (x) - 49 + 81

Add similar elements:

- cos^{2} (x) - 81 cos^{2} (x) = - 82 cos^{2} (x)

= - 82 cos^{2} (x) + 14 cos (x) - 49 + 81

Add/Subtract the numbers:

- 49 + 81 = 32

= 14 cos (x) - 82 cos^{2} (x) + 32

= 14 cos (x) - 82 cos^{2} (x) + 32

= 14 cos (x) - 82 cos^{2} (x) + 32

32 + 14 cos (x) - 82 cos^{2} (x) = 0

Solve by substitution

32 + 14 cos (x) - 82 cos^{2} (x) = 0

Let:

cos (x) = u

32 + 14 u - 82 u^{2} = 0

32 + 14 u - 82 u^{2} = 0 : u = - \frac{- 7 + 9 33}{82}, u = \frac{7 + 9 33}{82}

32 + 14 u - 82 u^{2} = 0

Write in the standard form

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

- 82 u^{2} + 14 u + 32 = 0

Solve with the quadratic formula

- 82 u^{2} + 14 u + 32 = 0

Quadratic Equation Formula:

For

a = - 82, b = 14, c = 32

u_{1, 2} = \frac{- 14 \pm 1 4 ^{2} - 4 ( - 82 ) \cdot 32}{2 ( - 82 )}

u_{1, 2} = \frac{- 14 \pm 1 4 ^{2} - 4 ( - 82 ) \cdot 32}{2 ( - 82 )}

1 4^{2} - 4 (- 82) \cdot 32 = 1833

1 4^{2} - 4 (- 82) \cdot 32

Apply rule

- (- a) = a

= 1 4^{2} + 4 \cdot 82 \cdot 32

Multiply the numbers:

4 \cdot 82 \cdot 32 = 10496

= 1 4^{2} + 10496

1 4^{2} = 196

= 196 + 10496

Add the numbers:

196 + 10496 = 10692

= 10692

Prime factorization of

10692 : 2^{2} \cdot 3^{5} \cdot 11

10692

10692

divides by

210692 = 5346 \cdot 2

= 2 \cdot 5346

5346

divides by

25346 = 2673 \cdot 2

= 2 \cdot 2 \cdot 2673

2673

divides by

32673 = 891 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 891

891

divides by

3891 = 297 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 297

297

divides by

3297 = 99 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 99

99

divides by

399 = 33 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 33

33

divides by

333 = 11 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 11

2, 3, 11

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 11

= 2^{2} \cdot 3^{5} \cdot 11

= 3^{5} \cdot 2^{2} \cdot 11

Apply exponent rule:

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

= 3^{4} \cdot 2^{2} \cdot 3 \cdot 11

Apply radical rule:

= 2^{2} 3^{4} 3 \cdot 11

Apply radical rule:

2^{2} = 2

= 2 3^{4} 3 \cdot 11

Apply radical rule:

3^{4} = 3^{\frac{4}{2}} = 3^{2}

= 3^{2} \cdot 2 3 \cdot 11

Refine

= 1833

u_{1, 2} = \frac{- 14 \pm 18 33}{2 ( - 82 )}

Separate the solutions

u_{1} = \frac{- 14 + 18 33}{2 ( - 82 )}, u_{2} = \frac{- 14 - 18 33}{2 ( - 82 )}

u = \frac{- 14 + 18 33}{2 ( - 82 )} : - \frac{- 7 + 9 33}{82}

\frac{- 14 + 18 33}{2 ( - 82 )}

Remove parentheses:

(- a) = - a

= \frac{- 14 + 18 33}{- 2 \cdot 82}

Multiply the numbers:

2 \cdot 82 = 164

= \frac{- 14 + 18 33}{- 164}

Apply the fraction rule:

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

= - \frac{- 14 + 18 33}{164}

Cancel

\frac{- 14 + 18 33}{164} : \frac{9 33 - 7}{82}

\frac{- 14 + 18 33}{164}

Factor

- 14 + 1833 : 2 (- 7 + 933)

- 14 + 1833

Rewrite as

= - 2 \cdot 7 + 2 \cdot 933

Factor out common term

2

= 2 (- 7 + 933)

= \frac{2 ( - 7 + 9 33 )}{164}

Cancel the common factor:

2

= \frac{- 7 + 9 33}{82}

= - \frac{9 33 - 7}{82}

= - \frac{- 7 + 9 33}{82}

u = \frac{- 14 - 18 33}{2 ( - 82 )} : \frac{7 + 9 33}{82}

\frac{- 14 - 18 33}{2 ( - 82 )}

Remove parentheses:

(- a) = - a

= \frac{- 14 - 18 33}{- 2 \cdot 82}

Multiply the numbers:

2 \cdot 82 = 164

= \frac{- 14 - 18 33}{- 164}

Apply the fraction rule:

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

- 14 - 1833 = - (14 + 1833)

= \frac{14 + 18 33}{164}

Factor

14 + 1833 : 2 (7 + 933)

14 + 1833

Rewrite as

= 2 \cdot 7 + 2 \cdot 933

Factor out common term

2

= 2 (7 + 933)

= \frac{2 ( 7 + 9 33 )}{164}

Cancel the common factor:

2

= \frac{7 + 9 33}{82}

The solutions to the quadratic equation are:

u = - \frac{- 7 + 9 33}{82}, u = \frac{7 + 9 33}{82}

Substitute back

u = cos (x)

cos (x) = - \frac{- 7 + 9 33}{82}, cos (x) = \frac{7 + 9 33}{82}

cos (x) = - \frac{- 7 + 9 33}{82}, cos (x) = \frac{7 + 9 33}{82}

cos (x) = - \frac{- 7 + 9 33}{82} : x = arccos (- \frac{- 7 + 9 33}{82}) + 2 πn, x = - arccos (- \frac{- 7 + 9 33}{82}) + 2 πn

cos (x) = - \frac{- 7 + 9 33}{82}

Apply trig inverse properties

cos (x) = - \frac{- 7 + 9 33}{82}

General solutions for

cos (x) = - \frac{- 7 + 9 33}{82}

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

x = arccos (- \frac{- 7 + 9 33}{82}) + 2 πn, x = - arccos (- \frac{- 7 + 9 33}{82}) + 2 πn

x = arccos (- \frac{- 7 + 9 33}{82}) + 2 πn, x = - arccos (- \frac{- 7 + 9 33}{82}) + 2 πn

cos (x) = \frac{7 + 9 33}{82} : x = arccos (\frac{7 + 9 33}{82}) + 2 πn, x = 2 π - arccos (\frac{7 + 9 33}{82}) + 2 πn

cos (x) = \frac{7 + 9 33}{82}

Apply trig inverse properties

cos (x) = \frac{7 + 9 33}{82}

General solutions for

cos (x) = \frac{7 + 9 33}{82}

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

x = arccos (\frac{7 + 9 33}{82}) + 2 πn, x = 2 π - arccos (\frac{7 + 9 33}{82}) + 2 πn

x = arccos (\frac{7 + 9 33}{82}) + 2 πn, x = 2 π - arccos (\frac{7 + 9 33}{82}) + 2 πn

Combine all the solutions

x = arccos (- \frac{- 7 + 9 33}{82}) + 2 πn, x = - arccos (- \frac{- 7 + 9 33}{82}) + 2 πn, x = arccos (\frac{7 + 9 33}{82}) + 2 πn, x = 2 π - arccos (\frac{7 + 9 33}{82}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

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

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

Check the solution

arccos (- \frac{- 7 + 9 33}{82}) + 2 πn :

False

arccos (- \frac{- 7 + 9 33}{82}) + 2 πn

Plug in

n = 1

arccos (- \frac{- 7 + 9 33}{82}) + 2 π 1

For

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

plug in

x = arccos (- \frac{- 7 + 9 33}{82}) + 2 π 1

9 sin (arccos (- \frac{- 7 + 9 33}{82}) + 2 π 1) = cos (arccos (- \frac{- 7 + 9 33}{82}) + 2 π 1) - 7

Refine

7.54513 \dots = - 7.54513 \dots

\Rightarrow False

Check the solution

- arccos (- \frac{- 7 + 9 33}{82}) + 2 πn :

True

- arccos (- \frac{- 7 + 9 33}{82}) + 2 πn

Plug in

n = 1

- arccos (- \frac{- 7 + 9 33}{82}) + 2 π 1

For

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

plug in

x = - arccos (- \frac{- 7 + 9 33}{82}) + 2 π 1

9 sin (- arccos (- \frac{- 7 + 9 33}{82}) + 2 π 1) = cos (- arccos (- \frac{- 7 + 9 33}{82}) + 2 π 1) - 7

Refine

- 7.54513 \dots = - 7.54513 \dots

\Rightarrow True

Check the solution

arccos (\frac{7 + 9 33}{82}) + 2 πn :

False

arccos (\frac{7 + 9 33}{82}) + 2 πn

Plug in

n = 1

arccos (\frac{7 + 9 33}{82}) + 2 π 1

For

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

plug in

x = arccos (\frac{7 + 9 33}{82}) + 2 π 1

9 sin (arccos (\frac{7 + 9 33}{82}) + 2 π 1) = cos (arccos (\frac{7 + 9 33}{82}) + 2 π 1) - 7

Refine

6.28413 \dots = - 6.28413 \dots

\Rightarrow False

Check the solution

2 π - arccos (\frac{7 + 9 33}{82}) + 2 πn :

True

2 π - arccos (\frac{7 + 9 33}{82}) + 2 πn

Plug in

n = 1

2 π - arccos (\frac{7 + 9 33}{82}) + 2 π 1

For

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

plug in

x = 2 π - arccos (\frac{7 + 9 33}{82}) + 2 π 1

9 sin (2 π - arccos (\frac{7 + 9 33}{82}) + 2 π 1) = cos (2 π - arccos (\frac{7 + 9 33}{82}) + 2 π 1) - 7

Refine

- 6.28413 \dots = - 6.28413 \dots

\Rightarrow True

x = - arccos (- \frac{- 7 + 9 33}{82}) + 2 πn, x = 2 π - arccos (\frac{7 + 9 33}{82}) + 2 πn

Show solutions in decimal form

x = - 2.14734 \dots + 2 πn, x = 2 π - 0.77293 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 9sin(x)=cos(x)-7 ?
The general solution for 9sin(x)=cos(x)-7 is x=-2.14734…+2pin,x=2pi-0.77293…+2pin