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Popular >

2cos(a)=sqrt(2)*(1-sin(x))

Solution

2 cos (a) = 2 \cdot (1 - sin (x))

Solution

x = arcsin (- 2 cos (a) + 1) + 2 πn, x = π + arcsin (2 cos (a) - 1) + 2 πn

Solution steps

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

Switch sides

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

Divide both sides by

2

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

Divide both sides by

2

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

Simplify

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

Simplify

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

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

Cancel the common factor:

2

= 1 - sin (x)

Simplify

\frac{2 cos ( a )}{2} : 2 cos (a)

\frac{2 cos ( a )}{2}

Apply radical rule:

n a = a^{\frac{1}{n}}

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

= \frac{2 cos ( a )}{2 ^{\frac{1}{2}}}

Apply exponent rule:

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

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

= 2^{- \frac{1}{2} + 1} cos (a)

Subtract the numbers:

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

= 2^{\frac{1}{2}} cos (a)

Apply radical rule:

a^{\frac{1}{n}} = n a

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

= 2 cos (a)

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

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

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

Move

1

to the right side

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

Subtract

1

from both sides

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

Simplify

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

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

Divide both sides by

- 1

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

Divide both sides by

- 1

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

Simplify

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

Simplify

\frac{- sin ( x )}{- 1} : sin (x)

\frac{- sin ( x )}{- 1}

Apply the fraction rule:

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

= \frac{sin ( x )}{1}

Apply rule

\frac{a}{1} = a

= sin (x)

Simplify

\frac{2 cos ( a )}{- 1} - \frac{1}{- 1} : - 2 cos (a) + 1

\frac{2 cos ( a )}{- 1} - \frac{1}{- 1}

Apply rule

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

= \frac{2 cos ( a ) - 1}{- 1}

Apply the fraction rule:

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

= - \frac{2 cos ( a ) - 1}{1}

Apply rule

\frac{a}{1} = a

= - (2 cos (a) - 1)

Distribute parentheses

= - (2 cos (a)) - (- 1)

Apply minus-plus rules

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

= - 2 cos (a) + 1

sin (x) = - 2 cos (a) + 1

sin (x) = - 2 cos (a) + 1

sin (x) = - 2 cos (a) + 1

Apply trig inverse properties

sin (x) = - 2 cos (a) + 1

General solutions for

sin (x) = - 2 cos (a) + 1

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

x = arcsin (- 2 cos (a) + 1) + 2 πn, x = π + arcsin (2 cos (a) - 1) + 2 πn

x = arcsin (- 2 cos (a) + 1) + 2 πn, x = π + arcsin (2 cos (a) - 1) + 2 πn

Graph

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