What is the general solution for cos(a)=(57)/(sqrt(113)*\sqrt{78)} ?

The general solution for cos(a)=(57)/(sqrt(113)*\sqrt{78)} is a=0.91834…+2pin,a=2pi-0.91834…+2pin

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cos(a)=(57)/(sqrt(113)*\sqrt{78)}

Solution

cos (a) = \frac{57}{113 \cdot 78}

Solution

a = 0.91834 \dots + 2 πn, a = 2 π - 0.91834 \dots + 2 πn

Radians

a = 0.91834 \dots + 6.28318 \dots n, a = 5.36484 \dots + 6.28318 \dots n

Solution steps

cos (a) = \frac{57}{113 78}

Simplify

\frac{57}{113 78} : \frac{19 8814}{2938}

\frac{57}{113 78}

Factor

57 : 3 \cdot 19

Factor

57 = 3 \cdot 19

Factor

78 : 2313

Factor

78 = 2 \cdot 3 \cdot 13

= 2 \cdot 3 \cdot 13

Apply radical rule:

= 2313

= \frac{3 \cdot 19}{113 2 3 13}

Cancel

\frac{3 \cdot 19}{113 2 3 13} : \frac{3 \cdot 19}{113 2 13}

\frac{3 \cdot 19}{113 2 3 13}

Apply radical rule:

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

= \frac{3 \cdot 19}{113 2 \cdot 3 ^{\frac{1}{2}} 13}

Apply exponent rule:

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

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

= \frac{19 \cdot 3 ^{- \frac{1}{2} + 1}}{113 2 13}

Subtract the numbers:

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

= \frac{19 \cdot 3 ^{\frac{1}{2}}}{113 2 13}

Apply radical rule:

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

= \frac{19 3}{113 2 13}

= \frac{3 \cdot 19}{113 2 13}

Simplify

113213 : 2938

113213

Apply radical rule:

a b = a \cdot b

113213 = 113 \cdot 2 \cdot 13

= 113 \cdot 2 \cdot 13

Multiply the numbers:

113 \cdot 2 \cdot 13 = 2938

= 2938

= \frac{19 3}{2938}

Rationalize

\frac{19 3}{2938} : \frac{19 8814}{2938}

\frac{19 3}{2938}

Multiply by the conjugate

\frac{2938}{2938}

= \frac{3 \cdot 19 2938}{2938 2938}

3 \cdot 192938 = 198814

3 \cdot 192938

Apply radical rule:

a b = a \cdot b

32938 = 3 \cdot 2938

= 19 3 \cdot 2938

Multiply the numbers:

3 \cdot 2938 = 8814

= 198814

29382938 = 2938

29382938

Apply radical rule:

a a = a

29382938 = 2938

= 2938

= \frac{19 8814}{2938}

= \frac{19 8814}{2938}

cos (a) = \frac{19 8814}{2938}

Apply trig inverse properties

cos (a) = \frac{19 8814}{2938}

General solutions for

cos (a) = \frac{19 8814}{2938}

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

a = arccos (\frac{19 8814}{2938}) + 2 πn, a = 2 π - arccos (\frac{19 8814}{2938}) + 2 πn

a = arccos (\frac{19 8814}{2938}) + 2 πn, a = 2 π - arccos (\frac{19 8814}{2938}) + 2 πn

Show solutions in decimal form

a = 0.91834 \dots + 2 πn, a = 2 π - 0.91834 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for cos(a)=(57)/(sqrt(113)*\sqrt{78)} ?
The general solution for cos(a)=(57)/(sqrt(113)*\sqrt{78)} is a=0.91834…+2pin,a=2pi-0.91834…+2pin