What is the general solution for cos(x)-sin(x)= 2/3 ?

The general solution for cos(x)-sin(x)= 2/3 is x=pi+1.27628…+2pin,x=0.29451…+2pin

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cos(x)-sin(x)= 2/3

Solution

cos (x) - sin (x) = \frac{2}{3}

Solution

x = π + 1.27628 \dots + 2 πn, x = 0.29451 \dots + 2 πn

Degrees

x = 253.12550 \dots^{\circ} + 36 0^{\circ} n, x = 16.87449 \dots^{\circ} + 36 0^{\circ} n

Solution steps

cos (x) - sin (x) = \frac{2}{3}

Add

sin (x)

to both sides

cos (x) = \frac{2}{3} + sin (x)

Square both sides

cos^{2} (x) = (\frac{2}{3} + sin (x))^{2}

Subtract

(\frac{2}{3} + sin (x))^{2}

from both sides

cos^{2} (x) - \frac{4}{9} - \frac{4}{3} sin (x) - sin^{2} (x) = 0

Simplify

cos^{2} (x) - \frac{4}{9} - \frac{4}{3} sin (x) - sin^{2} (x) : \frac{9 cos ^{2} ( x ) - 4 - 12 sin ( x ) - 9 sin ^{2} ( x )}{9}

cos^{2} (x) - \frac{4}{9} - \frac{4}{3} sin (x) - sin^{2} (x)

Multiply

\frac{4}{3} sin (x) : \frac{4 sin ( x )}{3}

\frac{4}{3} sin (x)

Multiply fractions:

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

= \frac{4 sin ( x )}{3}

= cos^{2} (x) - \frac{4}{9} - \frac{4 sin ( x )}{3} - sin^{2} (x)

Convert element to fraction:

cos^{2} (x) = \frac{cos ^{2} ( x )}{1}, sin^{2} (x) = \frac{sin ^{2} ( x )}{1}

= \frac{cos ^{2} ( x )}{1} - \frac{4}{9} - \frac{4 sin ( x )}{3} - \frac{sin ^{2} ( x )}{1}

Least Common Multiplier of

1, 9, 3, 1 : 9

1, 9, 3, 1

Least Common Multiplier (LCM)

Prime factorization of

1

Prime factorization of

9 : 3 \cdot 3

9

9

divides by

39 = 3 \cdot 3

= 3 \cdot 3

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

1

Compute a number comprised of factors that appear in at least one of the following:

1, 9, 3, 1

= 3 \cdot 3

Multiply the numbers:

3 \cdot 3 = 9

= 9

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

9

For

\frac{cos ^{2} ( x )}{1} :

multiply the denominator and numerator by

9

\frac{cos ^{2} ( x )}{1} = \frac{cos ^{2} ( x ) \cdot 9}{1 \cdot 9} = \frac{cos ^{2} ( x ) \cdot 9}{9}

For

\frac{4 sin ( x )}{3} :

multiply the denominator and numerator by

3

\frac{4 sin ( x )}{3} = \frac{4 sin ( x ) \cdot 3}{3 \cdot 3} = \frac{12 sin ( x )}{9}

For

\frac{sin ^{2} ( x )}{1} :

multiply the denominator and numerator by

9

\frac{sin ^{2} ( x )}{1} = \frac{sin ^{2} ( x ) \cdot 9}{1 \cdot 9} = \frac{sin ^{2} ( x ) \cdot 9}{9}

= \frac{cos ^{2} ( x ) \cdot 9}{9} - \frac{4}{9} - \frac{12 sin ( x )}{9} - \frac{sin ^{2} ( x ) \cdot 9}{9}

Since the denominators are equal, combine the fractions:

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

= \frac{cos ^{2} ( x ) \cdot 9 - 4 - 12 sin ( x ) - sin ^{2} ( x ) \cdot 9}{9}

\frac{9 cos ^{2} ( x ) - 4 - 12 sin ( x ) - 9 sin ^{2} ( x )}{9} = 0

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

9 cos^{2} (x) - 4 - 12 sin (x) - 9 sin^{2} (x) = 0

Rewrite using trig identities

- 4 - 12 sin (x) + 9 cos^{2} (x) - 9 sin^{2} (x)

Use the Pythagorean identity:

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

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

= - 4 - 12 sin (x) + 9 (1 - sin^{2} (x)) - 9 sin^{2} (x)

Simplify

- 4 - 12 sin (x) + 9 (1 - sin^{2} (x)) - 9 sin^{2} (x) : - 18 sin^{2} (x) - 12 sin (x) + 5

- 4 - 12 sin (x) + 9 (1 - sin^{2} (x)) - 9 sin^{2} (x)

Expand

9 (1 - sin^{2} (x)) : 9 - 9 sin^{2} (x)

9 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 9, b = 1, c = sin^{2} (x)

= 9 \cdot 1 - 9 sin^{2} (x)

Multiply the numbers:

9 \cdot 1 = 9

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

= - 4 - 12 sin (x) + 9 - 9 sin^{2} (x) - 9 sin^{2} (x)

Simplify

- 4 - 12 sin (x) + 9 - 9 sin^{2} (x) - 9 sin^{2} (x) : - 18 sin^{2} (x) - 12 sin (x) + 5

- 4 - 12 sin (x) + 9 - 9 sin^{2} (x) - 9 sin^{2} (x)

Add similar elements:

- 9 sin^{2} (x) - 9 sin^{2} (x) = - 18 sin^{2} (x)

= - 4 - 12 sin (x) + 9 - 18 sin^{2} (x)

Group like terms

= - 12 sin (x) - 18 sin^{2} (x) - 4 + 9

Add/Subtract the numbers:

- 4 + 9 = 5

= - 18 sin^{2} (x) - 12 sin (x) + 5

= - 18 sin^{2} (x) - 12 sin (x) + 5

= - 18 sin^{2} (x) - 12 sin (x) + 5

5 - 12 sin (x) - 18 sin^{2} (x) = 0

Solve by substitution

5 - 12 sin (x) - 18 sin^{2} (x) = 0

Let:

sin (x) = u

5 - 12 u - 18 u^{2} = 0

5 - 12 u - 18 u^{2} = 0 : u = - \frac{2 + 14}{6}, u = \frac{14 - 2}{6}

5 - 12 u - 18 u^{2} = 0

Write in the standard form

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

- 18 u^{2} - 12 u + 5 = 0

Solve with the quadratic formula

- 18 u^{2} - 12 u + 5 = 0

Quadratic Equation Formula:

For

a = - 18, b = - 12, c = 5

u_{1, 2} = \frac{- ( - 12 ) \pm ( - 12 ) ^{2} - 4 ( - 18 ) \cdot 5}{2 ( - 18 )}

u_{1, 2} = \frac{- ( - 12 ) \pm ( - 12 ) ^{2} - 4 ( - 18 ) \cdot 5}{2 ( - 18 )}

(- 12)^{2} - 4 (- 18) \cdot 5 = 614

(- 12)^{2} - 4 (- 18) \cdot 5

Apply rule

- (- a) = a

= (- 12)^{2} + 4 \cdot 18 \cdot 5

Apply exponent rule:

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

n

is even

(- 12)^{2} = 1 2^{2}

= 1 2^{2} + 4 \cdot 18 \cdot 5

Multiply the numbers:

4 \cdot 18 \cdot 5 = 360

= 1 2^{2} + 360

1 2^{2} = 144

= 144 + 360

Add the numbers:

144 + 360 = 504

= 504

Prime factorization of

504 : 2^{3} \cdot 3^{2} \cdot 7

504

504

divides by

2504 = 252 \cdot 2

= 2 \cdot 252

252

divides by

2252 = 126 \cdot 2

= 2 \cdot 2 \cdot 126

126

divides by

2126 = 63 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 63

63

divides by

363 = 21 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 21

21

divides by

321 = 7 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 7

2, 3, 7

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 7

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

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

Apply exponent rule:

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

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

Apply radical rule:

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

Apply radical rule:

2^{2} = 2

= 2 3^{2} 2 \cdot 7

Apply radical rule:

3^{2} = 3

= 2 \cdot 3 2 \cdot 7

Refine

= 614

u_{1, 2} = \frac{- ( - 12 ) \pm 6 14}{2 ( - 18 )}

Separate the solutions

u_{1} = \frac{- ( - 12 ) + 6 14}{2 ( - 18 )}, u_{2} = \frac{- ( - 12 ) - 6 14}{2 ( - 18 )}

u = \frac{- ( - 12 ) + 6 14}{2 ( - 18 )} : - \frac{2 + 14}{6}

\frac{- ( - 12 ) + 6 14}{2 ( - 18 )}

Remove parentheses:

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

= \frac{12 + 6 14}{- 2 \cdot 18}

Multiply the numbers:

2 \cdot 18 = 36

= \frac{12 + 6 14}{- 36}

Apply the fraction rule:

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

= - \frac{12 + 6 14}{36}

Cancel

\frac{12 + 6 14}{36} : \frac{2 + 14}{6}

\frac{12 + 6 14}{36}

Factor

12 + 614 : 6 (2 + 14)

12 + 614

Rewrite as

= 6 \cdot 2 + 614

Factor out common term

6

= 6 (2 + 14)

= \frac{6 ( 2 + 14 )}{36}

Cancel the common factor:

6

= \frac{2 + 14}{6}

= - \frac{2 + 14}{6}

u = \frac{- ( - 12 ) - 6 14}{2 ( - 18 )} : \frac{14 - 2}{6}

\frac{- ( - 12 ) - 6 14}{2 ( - 18 )}

Remove parentheses:

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

= \frac{12 - 6 14}{- 2 \cdot 18}

Multiply the numbers:

2 \cdot 18 = 36

= \frac{12 - 6 14}{- 36}

Apply the fraction rule:

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

12 - 614 = - (614 - 12)

= \frac{6 14 - 12}{36}

Factor

614 - 12 : 6 (14 - 2)

614 - 12

Rewrite as

= 614 - 6 \cdot 2

Factor out common term

6

= 6 (14 - 2)

= \frac{6 ( 14 - 2 )}{36}

Cancel the common factor:

6

= \frac{14 - 2}{6}

The solutions to the quadratic equation are:

u = - \frac{2 + 14}{6}, u = \frac{14 - 2}{6}

Substitute back

u = sin (x)

sin (x) = - \frac{2 + 14}{6}, sin (x) = \frac{14 - 2}{6}

sin (x) = - \frac{2 + 14}{6}, sin (x) = \frac{14 - 2}{6}

sin (x) = - \frac{2 + 14}{6} : x = arcsin (- \frac{2 + 14}{6}) + 2 πn, x = π + arcsin (\frac{2 + 14}{6}) + 2 πn

sin (x) = - \frac{2 + 14}{6}

Apply trig inverse properties

sin (x) = - \frac{2 + 14}{6}

General solutions for

sin (x) = - \frac{2 + 14}{6}

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (- \frac{2 + 14}{6}) + 2 πn, x = π + arcsin (\frac{2 + 14}{6}) + 2 πn

x = arcsin (- \frac{2 + 14}{6}) + 2 πn, x = π + arcsin (\frac{2 + 14}{6}) + 2 πn

sin (x) = \frac{14 - 2}{6} : x = arcsin (\frac{14 - 2}{6}) + 2 πn, x = π - arcsin (\frac{14 - 2}{6}) + 2 πn

sin (x) = \frac{14 - 2}{6}

Apply trig inverse properties

sin (x) = \frac{14 - 2}{6}

General solutions for

sin (x) = \frac{14 - 2}{6}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin (\frac{14 - 2}{6}) + 2 πn, x = π - arcsin (\frac{14 - 2}{6}) + 2 πn

x = arcsin (\frac{14 - 2}{6}) + 2 πn, x = π - arcsin (\frac{14 - 2}{6}) + 2 πn

Combine all the solutions

x = arcsin (- \frac{2 + 14}{6}) + 2 πn, x = π + arcsin (\frac{2 + 14}{6}) + 2 πn, x = arcsin (\frac{14 - 2}{6}) + 2 πn, x = π - arcsin (\frac{14 - 2}{6}) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

cos (x) - sin (x) = \frac{2}{3}

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

Check the solution

arcsin (- \frac{2 + 14}{6}) + 2 πn :

False

arcsin (- \frac{2 + 14}{6}) + 2 πn

Plug in

n = 1

arcsin (- \frac{2 + 14}{6}) + 2 π 1

For

cos (x) - sin (x) = \frac{2}{3}

plug in

x = arcsin (- \frac{2 + 14}{6}) + 2 π 1

cos (arcsin (- \frac{2 + 14}{6}) + 2 π 1) - sin (arcsin (- \frac{2 + 14}{6}) + 2 π 1) = \frac{2}{3}

Refine

1.24721 \dots = 0.66666 \dots

\Rightarrow False

Check the solution

π + arcsin (\frac{2 + 14}{6}) + 2 πn :

True

π + arcsin (\frac{2 + 14}{6}) + 2 πn

Plug in

n = 1

π + arcsin (\frac{2 + 14}{6}) + 2 π 1

For

cos (x) - sin (x) = \frac{2}{3}

plug in

x = π + arcsin (\frac{2 + 14}{6}) + 2 π 1

cos (π + arcsin (\frac{2 + 14}{6}) + 2 π 1) - sin (π + arcsin (\frac{2 + 14}{6}) + 2 π 1) = \frac{2}{3}

Refine

0.66666 \dots = 0.66666 \dots

\Rightarrow True

Check the solution

arcsin (\frac{14 - 2}{6}) + 2 πn :

True

arcsin (\frac{14 - 2}{6}) + 2 πn

Plug in

n = 1

arcsin (\frac{14 - 2}{6}) + 2 π 1

For

cos (x) - sin (x) = \frac{2}{3}

plug in

x = arcsin (\frac{14 - 2}{6}) + 2 π 1

cos (arcsin (\frac{14 - 2}{6}) + 2 π 1) - sin (arcsin (\frac{14 - 2}{6}) + 2 π 1) = \frac{2}{3}

Refine

0.66666 \dots = 0.66666 \dots

\Rightarrow True

Check the solution

π - arcsin (\frac{14 - 2}{6}) + 2 πn :

False

π - arcsin (\frac{14 - 2}{6}) + 2 πn

Plug in

n = 1

π - arcsin (\frac{14 - 2}{6}) + 2 π 1

For

cos (x) - sin (x) = \frac{2}{3}

plug in

x = π - arcsin (\frac{14 - 2}{6}) + 2 π 1

cos (π - arcsin (\frac{14 - 2}{6}) + 2 π 1) - sin (π - arcsin (\frac{14 - 2}{6}) + 2 π 1) = \frac{2}{3}

Refine

- 1.24721 \dots = 0.66666 \dots

\Rightarrow False

x = π + arcsin (\frac{2 + 14}{6}) + 2 πn, x = arcsin (\frac{14 - 2}{6}) + 2 πn

Show solutions in decimal form

x = π + 1.27628 \dots + 2 πn, x = 0.29451 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for cos(x)-sin(x)= 2/3 ?
The general solution for cos(x)-sin(x)= 2/3 is x=pi+1.27628…+2pin,x=0.29451…+2pin