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Popular >

solvefor x,c^2=a^2+b^2-2ab*cos(x)

Solution

solve for

x, c^{2} = a^{2} + b^{2} - 2 ab \cdot cos (x)

Solution

x = arccos (- \frac{c ^{2} - a ^{2} - b ^{2}}{2 ab}) + 2 πn, x = - arccos (- \frac{c ^{2} - a ^{2} - b ^{2}}{2 ab}) + 2 πn

Solution steps

c^{2} = a^{2} + b^{2} - 2 ab cos (x)

Switch sides

a^{2} + b^{2} - 2 ab cos (x) = c^{2}

Move

a^{2}

to the right side

a^{2} + b^{2} - 2 ab cos (x) = c^{2}

Subtract

a^{2}

from both sides

a^{2} + b^{2} - 2 ab cos (x) - a^{2} = c^{2} - a^{2}

Simplify

b^{2} - 2 ab cos (x) = c^{2} - a^{2}

b^{2} - 2 ab cos (x) = c^{2} - a^{2}

Move

b^{2}

to the right side

b^{2} - 2 ab cos (x) = c^{2} - a^{2}

Subtract

b^{2}

from both sides

b^{2} - 2 ab cos (x) - b^{2} = c^{2} - a^{2} - b^{2}

Simplify

- 2 ab cos (x) = c^{2} - a^{2} - b^{2}

- 2 ab cos (x) = c^{2} - a^{2} - b^{2}

Divide both sides by

- 2 ab; - 2 ab \neq = 0

- 2 ab cos (x) = c^{2} - a^{2} - b^{2}

Divide both sides by

- 2 ab; - 2 ab \neq = 0

\frac{- 2 ab cos ( x )}{- 2 ab} = \frac{c ^{2}}{- 2 ab} - \frac{a ^{2}}{- 2 ab} - \frac{b ^{2}}{- 2 ab}; - 2 ab \neq = 0

Simplify

\frac{- 2 ab cos ( x )}{- 2 ab} = \frac{c ^{2}}{- 2 ab} - \frac{a ^{2}}{- 2 ab} - \frac{b ^{2}}{- 2 ab}

Simplify

\frac{- 2 ab cos ( x )}{- 2 ab} : cos (x)

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

Apply the fraction rule:

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

= \frac{2 ab cos ( x )}{2 ab}

Divide the numbers:

\frac{2}{2} = 1

= \frac{ab cos ( x )}{ab}

Cancel the common factor:

a

= \frac{b cos ( x )}{b}

Cancel the common factor:

b

= cos (x)

Simplify

\frac{c ^{2}}{- 2 ab} - \frac{a ^{2}}{- 2 ab} - \frac{b ^{2}}{- 2 ab} : - \frac{c ^{2} - a ^{2} - b ^{2}}{2 ab}

\frac{c ^{2}}{- 2 ab} - \frac{a ^{2}}{- 2 ab} - \frac{b ^{2}}{- 2 ab}

Apply rule

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

= \frac{c ^{2} - a ^{2} - b ^{2}}{- 2 ab}

Apply the fraction rule:

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

= - \frac{c ^{2} - a ^{2} - b ^{2}}{2 ab}

cos (x) = - \frac{c ^{2} - a ^{2} - b ^{2}}{2 ab}; - 2 ab \neq = 0

cos (x) = - \frac{c ^{2} - a ^{2} - b ^{2}}{2 ab}; - 2 ab \neq = 0

cos (x) = - \frac{c ^{2} - a ^{2} - b ^{2}}{2 ab}; - 2 ab \neq = 0

Apply trig inverse properties

cos (x) = - \frac{c ^{2} - a ^{2} - b ^{2}}{2 ab}

General solutions for

cos (x) = - \frac{c ^{2} - a ^{2} - b ^{2}}{2 ab}

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

x = arccos (- \frac{c ^{2} - a ^{2} - b ^{2}}{2 ab}) + 2 πn, x = - arccos (- \frac{c ^{2} - a ^{2} - b ^{2}}{2 ab}) + 2 πn

x = arccos (- \frac{c ^{2} - a ^{2} - b ^{2}}{2 ab}) + 2 πn, x = - arccos (- \frac{c ^{2} - a ^{2} - b ^{2}}{2 ab}) + 2 πn

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