What is the general solution for sin(x)-cos(x)= 1/2 ?

The general solution for sin(x)-cos(x)= 1/2 is x=0.36136…+2pin+pi/4 ,x=pi-0.36136…+2pin+pi/4

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sin(x)-cos(x)= 1/2

Solution

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

Solution

x = 0.36136 \dots + 2 πn + \frac{π}{4}, x = π - 0.36136 \dots + 2 πn + \frac{π}{4}

Degrees

x = 65.70481 \dots^{\circ} + 36 0^{\circ} n, x = 204.29518 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Rewrite using trig identities

sin (x) - cos (x)

sin (x) - cos (x) = 2 sin (x - \frac{π}{4})

sin (x) - cos (x)

Rewrite as

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

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{1}{2}

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{1}{2}

= 2 (cos (\frac{π}{4}) sin (x) - sin (\frac{π}{4}) cos (x))

Use the Angle Sum identity:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

= 2 sin (x - \frac{π}{4})

= 2 sin (x - \frac{π}{4})

2 sin (x - \frac{π}{4}) = \frac{1}{2}

Divide both sides by

2

2 sin (x - \frac{π}{4}) = \frac{1}{2}

Divide both sides by

2

\frac{2 sin ( x - \frac{π}{4} )}{2} = \frac{\frac{1}{2}}{2}

Simplify

\frac{2 sin ( x - \frac{π}{4} )}{2} = \frac{\frac{1}{2}}{2}

Simplify

\frac{2 sin ( x - \frac{π}{4} )}{2} : sin (x - \frac{π}{4})

\frac{2 sin ( x - \frac{π}{4} )}{2}

Cancel the common factor:

2

= sin (x - \frac{π}{4})

Simplify

\frac{\frac{1}{2}}{2} : \frac{2}{4}

\frac{\frac{1}{2}}{2}

Apply the fraction rule:

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

= \frac{1}{2 2}

Rationalize

\frac{1}{2 2} : \frac{2}{4}

\frac{1}{2 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{1 \cdot 2}{2 2 2}

1 \cdot 2 = 2

222 = 4

222

Apply exponent rule:

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

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 2^{1 + \frac{1}{2} + \frac{1}{2}}

Add similar elements:

\frac{1}{2} + \frac{1}{2} = 2 \cdot \frac{1}{2}

= 2^{1 + 2 \cdot \frac{1}{2}}

2 \cdot \frac{1}{2} = 1

2 \cdot \frac{1}{2}

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2}{4}

= \frac{2}{4}

sin (x - \frac{π}{4}) = \frac{2}{4}

sin (x - \frac{π}{4}) = \frac{2}{4}

sin (x - \frac{π}{4}) = \frac{2}{4}

Apply trig inverse properties

sin (x - \frac{π}{4}) = \frac{2}{4}

General solutions for

sin (x - \frac{π}{4}) = \frac{2}{4}

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

x - \frac{π}{4} = arcsin (\frac{2}{4}) + 2 πn, x - \frac{π}{4} = π - arcsin (\frac{2}{4}) + 2 πn

x - \frac{π}{4} = arcsin (\frac{2}{4}) + 2 πn, x - \frac{π}{4} = π - arcsin (\frac{2}{4}) + 2 πn

Solve

x - \frac{π}{4} = arcsin (\frac{2}{4}) + 2 πn : x = arcsin (\frac{1}{2 2}) + 2 πn + \frac{π}{4}

x - \frac{π}{4} = arcsin (\frac{2}{4}) + 2 πn

Simplify

arcsin (\frac{2}{4}) + 2 πn : arcsin (\frac{1}{2 2}) + 2 πn

arcsin (\frac{2}{4}) + 2 πn

\frac{2}{4} = \frac{1}{2 2}

\frac{2}{4}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

= \frac{2}{2 ^{2}}

Cancel

\frac{2}{2 ^{2}} : \frac{1}{2 ^{\frac{3}{2}}}

\frac{2}{2 ^{2}}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2 ^{\frac{1}{2}}}{2 ^{2}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{2}} = \frac{1}{2 ^{2 - \frac{1}{2}}}

= \frac{1}{2 ^{2 - \frac{1}{2}}}

Subtract the numbers:

2 - \frac{1}{2} = \frac{3}{2}

= \frac{1}{2 ^{\frac{3}{2}}}

= \frac{1}{2 ^{\frac{3}{2}}}

2^{\frac{3}{2}} = 22

2^{\frac{3}{2}}

2^{\frac{3}{2}} = 2^{1 + \frac{1}{2}}

= 2^{1 + \frac{1}{2}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

= 2^{1} \cdot 2^{\frac{1}{2}}

Refine

= 22

= \frac{1}{2 2}

= arcsin (\frac{1}{2 2}) + 2 πn

x - \frac{π}{4} = arcsin (\frac{1}{2 2}) + 2 πn

Move

\frac{π}{4}

to the right side

x - \frac{π}{4} = arcsin (\frac{1}{2 2}) + 2 πn

Add

\frac{π}{4}

to both sides

x - \frac{π}{4} + \frac{π}{4} = arcsin (\frac{1}{2 2}) + 2 πn + \frac{π}{4}

Simplify

x = arcsin (\frac{1}{2 2}) + 2 πn + \frac{π}{4}

x = arcsin (\frac{1}{2 2}) + 2 πn + \frac{π}{4}

Solve

x - \frac{π}{4} = π - arcsin (\frac{2}{4}) + 2 πn : x = π - arcsin (\frac{1}{2 2}) + 2 πn + \frac{π}{4}

x - \frac{π}{4} = π - arcsin (\frac{2}{4}) + 2 πn

Simplify

π - arcsin (\frac{2}{4}) + 2 πn : π - arcsin (\frac{1}{2 2}) + 2 πn

π - arcsin (\frac{2}{4}) + 2 πn

\frac{2}{4} = \frac{1}{2 2}

\frac{2}{4}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

= \frac{2}{2 ^{2}}

Cancel

\frac{2}{2 ^{2}} : \frac{1}{2 ^{\frac{3}{2}}}

\frac{2}{2 ^{2}}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2 ^{\frac{1}{2}}}{2 ^{2}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{2}} = \frac{1}{2 ^{2 - \frac{1}{2}}}

= \frac{1}{2 ^{2 - \frac{1}{2}}}

Subtract the numbers:

2 - \frac{1}{2} = \frac{3}{2}

= \frac{1}{2 ^{\frac{3}{2}}}

= \frac{1}{2 ^{\frac{3}{2}}}

2^{\frac{3}{2}} = 22

2^{\frac{3}{2}}

2^{\frac{3}{2}} = 2^{1 + \frac{1}{2}}

= 2^{1 + \frac{1}{2}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

= 2^{1} \cdot 2^{\frac{1}{2}}

Refine

= 22

= \frac{1}{2 2}

= π - arcsin (\frac{1}{2 2}) + 2 πn

x - \frac{π}{4} = π - arcsin (\frac{1}{2 2}) + 2 πn

Move

\frac{π}{4}

to the right side

x - \frac{π}{4} = π - arcsin (\frac{1}{2 2}) + 2 πn

Add

\frac{π}{4}

to both sides

x - \frac{π}{4} + \frac{π}{4} = π - arcsin (\frac{1}{2 2}) + 2 πn + \frac{π}{4}

Simplify

x = π - arcsin (\frac{1}{2 2}) + 2 πn + \frac{π}{4}

x = π - arcsin (\frac{1}{2 2}) + 2 πn + \frac{π}{4}

x = arcsin (\frac{1}{2 2}) + 2 πn + \frac{π}{4}, x = π - arcsin (\frac{1}{2 2}) + 2 πn + \frac{π}{4}

Show solutions in decimal form

x = 0.36136 \dots + 2 πn + \frac{π}{4}, x = π - 0.36136 \dots + 2 πn + \frac{π}{4}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x)-cos(x)= 1/2 ?
The general solution for sin(x)-cos(x)= 1/2 is x=0.36136…+2pin+pi/4 ,x=pi-0.36136…+2pin+pi/4