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solvefor b,a=arccos((a^2-b^2-c^2)/(-2bc))

Solution

solve for

b, a = arccos (\frac{a ^{2} - b ^{2} - c ^{2}}{- 2 b c})

Solution

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

Solution steps

a = arccos (\frac{a ^{2} - b ^{2} - c ^{2}}{- 2 b c})

Switch sides

arccos (\frac{a ^{2} - b ^{2} - c ^{2}}{- 2 b c}) = a

Apply trig inverse properties

arccos (\frac{a ^{2} - b ^{2} - c ^{2}}{- 2 b c}) = a

arccos (x) = a \Rightarrow x = cos (a)

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

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

Solve

\frac{a ^{2} - b ^{2} - c ^{2}}{- 2 b c} = cos (a) : b = c cos (a) + c^{2} cos^{2} (a) - c^{2} + a^{2}, b = c cos (a) - c^{2} cos^{2} (a) - c^{2} + a^{2}; c \neq = 0

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

Simplify

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

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

Apply the fraction rule:

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

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

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

Multiply both sides by

b

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

Multiply both sides by

b

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

Simplify

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

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

Solve

- \frac{a ^{2} - b ^{2} - c ^{2}}{2 c} = cos (a) b : b = c cos (a) + c^{2} cos^{2} (a) - c^{2} + a^{2}, b = c cos (a) - c^{2} cos^{2} (a) - c^{2} + a^{2}

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

Multiply both sides by

2 c

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

Multiply both sides by

2 c

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

Simplify

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

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

Expand

- (a^{2} - b^{2} - c^{2}) : - a^{2} + b^{2} + c^{2}

- (a^{2} - b^{2} - c^{2})

Distribute parentheses

= - a^{2} - (- b^{2}) - (- c^{2})

Apply minus-plus rules

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

= - a^{2} + b^{2} + c^{2}

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

Move

2 b c cos (a)

to the left side

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

Subtract

2 b c cos (a)

from both sides

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

Simplify

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

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

Write in the standard form

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

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

Solve with the quadratic formula

b^{2} - 2 c cos (a) b - a^{2} + c^{2} = 0

Quadratic Equation Formula:

For

a = 1, b = - 2 c cos (a), c = - a^{2} + c^{2}

b_{1, 2} = \frac{- ( - 2 c cos ( a ) ) \pm ( - 2 c cos ( a ) ) ^{2} - 4 \cdot 1 \cdot ( - a ^{2} + c ^{2} )}{2 \cdot 1}

b_{1, 2} = \frac{- ( - 2 c cos ( a ) ) \pm ( - 2 c cos ( a ) ) ^{2} - 4 \cdot 1 \cdot ( - a ^{2} + c ^{2} )}{2 \cdot 1}

Simplify

(- 2 c cos (a))^{2} - 4 \cdot 1 \cdot (- a^{2} + c^{2}) : 2 c^{2} cos^{2} (a) + a^{2} - c^{2}

(- 2 c cos (a))^{2} - 4 \cdot 1 \cdot (- a^{2} + c^{2})

(- 2 c cos (a))^{2} = 2^{2} c^{2} cos^{2} (a)

(- 2 c cos (a))^{2}

Apply exponent rule:

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

n

is even

(- 2 c cos (a))^{2} = (2 c cos (a))^{2}

= (2 c cos (a))^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{2} c^{2} cos^{2} (a)

4 \cdot 1 \cdot (- a^{2} + c^{2}) = 4 (- a^{2} + c^{2})

4 \cdot 1 \cdot (- a^{2} + c^{2})

Multiply the numbers:

4 \cdot 1 = 4

= 4 (c^{2} - a^{2})

= 2^{2} c^{2} cos^{2} (a) - 4 (c^{2} - a^{2})

Factor

2^{2} c^{2} cos^{2} (a) - 4 (- a^{2} + c^{2}) : 4 (c^{2} cos^{2} (a) + a^{2} - c^{2})

2^{2} c^{2} cos^{2} (a) - 4 (- a^{2} + c^{2})

Rewrite as

= 4 c^{2} cos^{2} (a) - 4 (- a^{2} + c^{2})

Factor out common term

4

= 4 (c^{2} cos^{2} (a) - (- a^{2} + c^{2}))

Expand

c^{2} cos^{2} (a) - (c^{2} - a^{2}) : c^{2} cos^{2} (a) + a^{2} - c^{2}

c^{2} cos^{2} (a) - (- a^{2} + c^{2})

- (- a^{2} + c^{2}) : a^{2} - c^{2}

- (- a^{2} + c^{2})

Distribute parentheses

= - (- a^{2}) - c^{2}

Apply minus-plus rules

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

= a^{2} - c^{2}

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

= 4 (c^{2} cos^{2} (a) - c^{2} + a^{2})

= 4 (c^{2} cos^{2} (a) + a^{2} - c^{2})

Apply radical rule:

assuming

a \geq 0, b \geq 0

= 4 c^{2} cos^{2} (a) - c^{2} + a^{2}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

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

b_{1, 2} = \frac{- ( - 2 c cos ( a ) ) \pm 2 c ^{2} cos ^{2} ( a ) + a ^{2} - c ^{2}}{2 \cdot 1}

Separate the solutions

b_{1} = \frac{- ( - 2 c cos ( a ) ) + 2 c ^{2} cos ^{2} ( a ) + a ^{2} - c ^{2}}{2 \cdot 1}, b_{2} = \frac{- ( - 2 c cos ( a ) ) - 2 c ^{2} cos ^{2} ( a ) + a ^{2} - c ^{2}}{2 \cdot 1}

b = \frac{- ( - 2 c cos ( a ) ) + 2 c ^{2} cos ^{2} ( a ) + a ^{2} - c ^{2}}{2 \cdot 1} : c cos (a) + c^{2} cos^{2} (a) - c^{2} + a^{2}

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

Apply rule

- (- a) = a

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

Multiply the numbers:

2 \cdot 1 = 2

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

Factor out common term

2

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

Divide the numbers:

\frac{2}{2} = 1

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

b = \frac{- ( - 2 c cos ( a ) ) - 2 c ^{2} cos ^{2} ( a ) + a ^{2} - c ^{2}}{2 \cdot 1} : c cos (a) - c^{2} cos^{2} (a) - c^{2} + a^{2}

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

Apply rule

- (- a) = a

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

Multiply the numbers:

2 \cdot 1 = 2

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

Factor out common term

2

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

Divide the numbers:

\frac{2}{2} = 1

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

The solutions to the quadratic equation are:

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

b = c cos (a) + c^{2} cos^{2} (a) - c^{2} + a^{2}, b = c cos (a) - c^{2} cos^{2} (a) - c^{2} + a^{2}; c \neq = 0

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

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