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Popular >

-cos(x/2)+sin(x/2)=sqrt(2/5)

Solution

- cos (\frac{x}{2}) + sin (\frac{x}{2}) = \frac{2}{5}

Solution

x = 2 \cdot 0.46364 \dots + 4 πn + \frac{π}{2}, x = 2 π - 2 \cdot 0.46364 \dots + 4 πn + \frac{π}{2}

Degrees

x = 143.13010 \dots^{\circ} + 72 0^{\circ} n, x = 396.86989 \dots^{\circ} + 72 0^{\circ} n

Solution steps

- cos (\frac{x}{2}) + sin (\frac{x}{2}) = \frac{2}{5}

Rewrite using trig identities

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

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

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

Rewrite as

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

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

Use the Angle Sum identity:

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

= 2 sin (\frac{x}{2} - \frac{π}{4})

= 2 sin (\frac{x}{2} - \frac{π}{4})

2 sin (\frac{x}{2} - \frac{π}{4}) = \frac{2}{5}

Divide both sides by

2

2 sin (\frac{x}{2} - \frac{π}{4}) = \frac{2}{5}

Divide both sides by

2

\frac{2 sin ( \frac{x}{2} - \frac{π}{4} )}{2} = \frac{\frac{2}{5}}{2}

Simplify

\frac{2 sin ( \frac{x}{2} - \frac{π}{4} )}{2} = \frac{\frac{2}{5}}{2}

Simplify

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

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

Cancel the common factor:

2

= sin (\frac{x}{2} - \frac{π}{4})

Simplify

\frac{\frac{2}{5}}{2} : \frac{5}{5}

\frac{\frac{2}{5}}{2}

\frac{2}{5} = \frac{2}{5}

\frac{2}{5}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{2}{5}

= \frac{\frac{2}{5}}{2}

Apply the fraction rule:

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

= \frac{2}{5 2}

Cancel the common factor:

2

= \frac{1}{5}

Rationalize

\frac{1}{5} : \frac{5}{5}

\frac{1}{5}

Multiply by the conjugate

\frac{5}{5}

= \frac{1 \cdot 5}{5 5}

1 \cdot 5 = 5

55 = 5

55

Apply radical rule:

a a = a

55 = 5

= 5

= \frac{5}{5}

= \frac{5}{5}

sin (\frac{x}{2} - \frac{π}{4}) = \frac{5}{5}

sin (\frac{x}{2} - \frac{π}{4}) = \frac{5}{5}

sin (\frac{x}{2} - \frac{π}{4}) = \frac{5}{5}

Apply trig inverse properties

sin (\frac{x}{2} - \frac{π}{4}) = \frac{5}{5}

General solutions for

sin (\frac{x}{2} - \frac{π}{4}) = \frac{5}{5}

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

\frac{x}{2} - \frac{π}{4} = arcsin (\frac{5}{5}) + 2 πn, \frac{x}{2} - \frac{π}{4} = π - arcsin (\frac{5}{5}) + 2 πn

\frac{x}{2} - \frac{π}{4} = arcsin (\frac{5}{5}) + 2 πn, \frac{x}{2} - \frac{π}{4} = π - arcsin (\frac{5}{5}) + 2 πn

Solve

\frac{x}{2} - \frac{π}{4} = arcsin (\frac{5}{5}) + 2 πn : x = 2 arcsin (\frac{1}{5}) + 4 πn + \frac{π}{2}

\frac{x}{2} - \frac{π}{4} = arcsin (\frac{5}{5}) + 2 πn

Simplify

arcsin (\frac{5}{5}) + 2 πn : arcsin (\frac{1}{5}) + 2 πn

arcsin (\frac{5}{5}) + 2 πn

\frac{5}{5} = \frac{1}{5}

\frac{5}{5}

Apply radical rule:

5 = 5^{\frac{1}{2}}

= \frac{5 ^{\frac{1}{2}}}{5}

Apply exponent rule:

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

\frac{5 ^{\frac{1}{2}}}{5 ^{1}} = \frac{1}{5 ^{1 - \frac{1}{2}}}

= \frac{1}{5 ^{1 - \frac{1}{2}}}

Subtract the numbers:

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

= \frac{1}{5 ^{\frac{1}{2}}}

Apply radical rule:

5^{\frac{1}{2}} = 5

= \frac{1}{5}

= arcsin (\frac{1}{5}) + 2 πn

\frac{x}{2} - \frac{π}{4} = arcsin (\frac{1}{5}) + 2 πn

Move

\frac{π}{4}

to the right side

\frac{x}{2} - \frac{π}{4} = arcsin (\frac{1}{5}) + 2 πn

Add

\frac{π}{4}

to both sides

\frac{x}{2} - \frac{π}{4} + \frac{π}{4} = arcsin (\frac{1}{5}) + 2 πn + \frac{π}{4}

Simplify

\frac{x}{2} = arcsin (\frac{1}{5}) + 2 πn + \frac{π}{4}

\frac{x}{2} = arcsin (\frac{1}{5}) + 2 πn + \frac{π}{4}

Multiply both sides by

2

\frac{x}{2} = arcsin (\frac{1}{5}) + 2 πn + \frac{π}{4}

Multiply both sides by

2

\frac{2 x}{2} = 2 arcsin (\frac{1}{5}) + 2 \cdot 2 πn + 2 \cdot \frac{π}{4}

Simplify

\frac{2 x}{2} = 2 arcsin (\frac{1}{5}) + 2 \cdot 2 πn + 2 \cdot \frac{π}{4}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 arcsin (\frac{1}{5}) + 2 \cdot 2 πn + 2 \cdot \frac{π}{4} : 2 arcsin (\frac{1}{5}) + 4 πn + \frac{π}{2}

2 arcsin (\frac{1}{5}) + 2 \cdot 2 πn + 2 \cdot \frac{π}{4}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

2 \cdot \frac{π}{4} = \frac{π}{2}

2 \cdot \frac{π}{4}

Multiply fractions:

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

= \frac{π 2}{4}

Cancel the common factor:

2

= \frac{π}{2}

= 2 arcsin (\frac{1}{5}) + 4 πn + \frac{π}{2}

x = 2 arcsin (\frac{1}{5}) + 4 πn + \frac{π}{2}

x = 2 arcsin (\frac{1}{5}) + 4 πn + \frac{π}{2}

x = 2 arcsin (\frac{1}{5}) + 4 πn + \frac{π}{2}

Solve

\frac{x}{2} - \frac{π}{4} = π - arcsin (\frac{5}{5}) + 2 πn : x = 2 π - 2 arcsin (\frac{1}{5}) + 4 πn + \frac{π}{2}

\frac{x}{2} - \frac{π}{4} = π - arcsin (\frac{5}{5}) + 2 πn

Simplify

π - arcsin (\frac{5}{5}) + 2 πn : π - arcsin (\frac{1}{5}) + 2 πn

π - arcsin (\frac{5}{5}) + 2 πn

\frac{5}{5} = \frac{1}{5}

\frac{5}{5}

Apply radical rule:

5 = 5^{\frac{1}{2}}

= \frac{5 ^{\frac{1}{2}}}{5}

Apply exponent rule:

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

\frac{5 ^{\frac{1}{2}}}{5 ^{1}} = \frac{1}{5 ^{1 - \frac{1}{2}}}

= \frac{1}{5 ^{1 - \frac{1}{2}}}

Subtract the numbers:

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

= \frac{1}{5 ^{\frac{1}{2}}}

Apply radical rule:

5^{\frac{1}{2}} = 5

= \frac{1}{5}

= π - arcsin (\frac{1}{5}) + 2 πn

\frac{x}{2} - \frac{π}{4} = π - arcsin (\frac{1}{5}) + 2 πn

Move

\frac{π}{4}

to the right side

\frac{x}{2} - \frac{π}{4} = π - arcsin (\frac{1}{5}) + 2 πn

Add

\frac{π}{4}

to both sides

\frac{x}{2} - \frac{π}{4} + \frac{π}{4} = π - arcsin (\frac{1}{5}) + 2 πn + \frac{π}{4}

Simplify

\frac{x}{2} = π - arcsin (\frac{1}{5}) + 2 πn + \frac{π}{4}

\frac{x}{2} = π - arcsin (\frac{1}{5}) + 2 πn + \frac{π}{4}

Multiply both sides by

2

\frac{x}{2} = π - arcsin (\frac{1}{5}) + 2 πn + \frac{π}{4}

Multiply both sides by

2

\frac{2 x}{2} = 2 π - 2 arcsin (\frac{1}{5}) + 2 \cdot 2 πn + 2 \cdot \frac{π}{4}

Simplify

\frac{2 x}{2} = 2 π - 2 arcsin (\frac{1}{5}) + 2 \cdot 2 πn + 2 \cdot \frac{π}{4}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 π - 2 arcsin (\frac{1}{5}) + 2 \cdot 2 πn + 2 \cdot \frac{π}{4} : 2 π - 2 arcsin (\frac{1}{5}) + 4 πn + \frac{π}{2}

2 π - 2 arcsin (\frac{1}{5}) + 2 \cdot 2 πn + 2 \cdot \frac{π}{4}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

2 \cdot \frac{π}{4} = \frac{π}{2}

2 \cdot \frac{π}{4}

Multiply fractions:

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

= \frac{π 2}{4}

Cancel the common factor:

2

= \frac{π}{2}

= 2 π - 2 arcsin (\frac{1}{5}) + 4 πn + \frac{π}{2}

x = 2 π - 2 arcsin (\frac{1}{5}) + 4 πn + \frac{π}{2}

x = 2 π - 2 arcsin (\frac{1}{5}) + 4 πn + \frac{π}{2}

x = 2 π - 2 arcsin (\frac{1}{5}) + 4 πn + \frac{π}{2}

x = 2 arcsin (\frac{1}{5}) + 4 πn + \frac{π}{2}, x = 2 π - 2 arcsin (\frac{1}{5}) + 4 πn + \frac{π}{2}

Show solutions in decimal form

x = 2 \cdot 0.46364 \dots + 4 πn + \frac{π}{2}, x = 2 π - 2 \cdot 0.46364 \dots + 4 πn + \frac{π}{2}

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