What is the general solution for (cos(-30+x))/(cos(30+x))= 1835/726 ?

The general solution for (cos(-30+x))/(cos(30+x))= 1835/726 is x=0.64352…+180n

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(cos(-30+x))/(cos(30+x))= 1835/726

Solution

\frac{cos ( - 3 0 ^{\circ} + x )}{cos ( 3 0 ^{\circ} + x )} = \frac{1835}{726}

Solution

x = 0.64352 \dots + 18 0^{\circ} n

Radians

x = 0.64352 \dots + πn

Solution steps

\frac{cos ( - 3 0 ^{\circ} + x )}{cos ( 3 0 ^{\circ} + x )} = \frac{1835}{726}

Rewrite using trig identities

\frac{cos ( - 3 0 ^{\circ} + x )}{cos ( 3 0 ^{\circ} + x )} = \frac{1835}{726}

Rewrite using trig identities

cos (3 0^{\circ} + x)

Use the Angle Sum identity:

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

= cos (3 0^{\circ}) cos (x) - sin (3 0^{\circ}) sin (x)

Simplify

cos (3 0^{\circ}) cos (x) - sin (3 0^{\circ}) sin (x) : \frac{3}{2} cos (x) - \frac{1}{2} sin (x)

cos (3 0^{\circ}) cos (x) - sin (3 0^{\circ}) sin (x)

Simplify

cos (3 0^{\circ}) : \frac{3}{2}

cos (3 0^{\circ})

Use the following trivial identity:

cos (3 0^{\circ}) = \frac{3}{2}

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{3}{2}

= \frac{3}{2} cos (x) - sin (3 0^{\circ}) sin (x)

Simplify

sin (3 0^{\circ}) : \frac{1}{2}

sin (3 0^{\circ})

Use the following trivial identity:

sin (3 0^{\circ}) = \frac{1}{2}

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{1}{2}

= \frac{3}{2} cos (x) - \frac{1}{2} sin (x)

= \frac{3}{2} cos (x) - \frac{1}{2} sin (x)

Use the Angle Difference identity:

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

= cos (x) cos (3 0^{\circ}) + sin (x) sin (3 0^{\circ})

Simplify

cos (x) cos (3 0^{\circ}) + sin (x) sin (3 0^{\circ}) : \frac{3}{2} cos (x) + \frac{1}{2} sin (x)

cos (x) cos (3 0^{\circ}) + sin (x) sin (3 0^{\circ})

Simplify

cos (3 0^{\circ}) : \frac{3}{2}

cos (3 0^{\circ})

Use the following trivial identity:

cos (3 0^{\circ}) = \frac{3}{2}

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{3}{2}

= \frac{3}{2} cos (x) + sin (3 0^{\circ}) sin (x)

Simplify

sin (3 0^{\circ}) : \frac{1}{2}

sin (3 0^{\circ})

Use the following trivial identity:

sin (3 0^{\circ}) = \frac{1}{2}

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{1}{2}

= \frac{3}{2} cos (x) + \frac{1}{2} sin (x)

= \frac{3}{2} cos (x) + \frac{1}{2} sin (x)

\frac{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )}{\frac{3}{2} cos ( x ) - \frac{1}{2} sin ( x )} = \frac{1835}{726}

\frac{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )}{\frac{3}{2} cos ( x ) - \frac{1}{2} sin ( x )} = \frac{1835}{726}

Subtract

\frac{1835}{726}

from both sides

\frac{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )}{\frac{3}{2} cos ( x ) - \frac{1}{2} sin ( x )} - \frac{1835}{726} = 0

Simplify

\frac{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )}{\frac{3}{2} cos ( x ) - \frac{1}{2} sin ( x )} - \frac{1835}{726} : \frac{- 1109 3 cos ( x ) + 2561 sin ( x )}{726 ( 3 cos ( x ) - sin ( x ) )}

\frac{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )}{\frac{3}{2} cos ( x ) - \frac{1}{2} sin ( x )} - \frac{1835}{726}

\frac{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )}{\frac{3}{2} cos ( x ) - \frac{1}{2} sin ( x )} = \frac{3 cos ( x ) + sin ( x )}{3 cos ( x ) - sin ( x )}

\frac{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )}{\frac{3}{2} cos ( x ) - \frac{1}{2} sin ( x )}

\frac{3}{2} cos (x) = \frac{3 cos ( x )}{2}

\frac{3}{2} cos (x)

Multiply fractions:

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

= \frac{3 cos ( x )}{2}

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

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

Multiply fractions:

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

= \frac{1 \cdot sin ( x )}{2}

Multiply:

1 \cdot sin (x) = sin (x)

= \frac{sin ( x )}{2}

= \frac{\frac{3}{2} cos ( x ) + \frac{1}{2} sin ( x )}{\frac{3 c o s ( x )}{2} - \frac{s i n ( x )}{2}}

\frac{3}{2} cos (x) = \frac{3 cos ( x )}{2}

\frac{3}{2} cos (x)

Multiply fractions:

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

= \frac{3 cos ( x )}{2}

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

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

Multiply fractions:

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

= \frac{1 \cdot sin ( x )}{2}

Multiply:

1 \cdot sin (x) = sin (x)

= \frac{sin ( x )}{2}

= \frac{\frac{3 c o s ( x )}{2} + \frac{s i n ( x )}{2}}{\frac{3 c o s ( x )}{2} - \frac{s i n ( x )}{2}}

Combine the fractions

\frac{3 cos ( x )}{2} - \frac{sin ( x )}{2} : \frac{3 cos ( x ) - sin ( x )}{2}

Apply rule

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

= \frac{3 cos ( x ) - sin ( x )}{2}

= \frac{\frac{3 c o s ( x )}{2} + \frac{s i n ( x )}{2}}{\frac{3 c o s ( x ) - s i n ( x )}{2}}

Combine the fractions

\frac{3 cos ( x )}{2} + \frac{sin ( x )}{2} : \frac{3 cos ( x ) + sin ( x )}{2}

Apply rule

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

= \frac{3 cos ( x ) + sin ( x )}{2}

= \frac{\frac{3 c o s ( x ) + s i n ( x )}{2}}{\frac{3 c o s ( x ) - s i n ( x )}{2}}

Divide fractions:

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

= \frac{( 3 cos ( x ) + sin ( x ) ) \cdot 2}{2 ( 3 cos ( x ) - sin ( x ) )}

Cancel the common factor:

2

= \frac{3 cos ( x ) + sin ( x )}{3 cos ( x ) - sin ( x )}

= \frac{3 cos ( x ) + sin ( x )}{3 cos ( x ) - sin ( x )} - \frac{1835}{726}

Least Common Multiplier of

3 cos (x) - sin (x), 726 : 726 (3 cos (x) - sin (x))

3 cos (x) - sin (x), 726

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

3 cos (x) - sin (x)

726

= 726 (3 cos (x) - sin (x))

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

726 (3 cos (x) - sin (x))

For

\frac{3 cos ( x ) + sin ( x )}{3 cos ( x ) - sin ( x )} :

multiply the denominator and numerator by

726

\frac{3 cos ( x ) + sin ( x )}{3 cos ( x ) - sin ( x )} = \frac{( 3 cos ( x ) + sin ( x ) ) \cdot 726}{( 3 cos ( x ) - sin ( x ) ) \cdot 726}

For

\frac{1835}{726} :

multiply the denominator and numerator by

3 cos (x) - sin (x)

\frac{1835}{726} = \frac{1835 ( 3 cos ( x ) - sin ( x ) )}{726 ( 3 cos ( x ) - sin ( x ) )}

= \frac{( 3 cos ( x ) + sin ( x ) ) \cdot 726}{( 3 cos ( x ) - sin ( x ) ) \cdot 726} - \frac{1835 ( 3 cos ( x ) - sin ( x ) )}{726 ( 3 cos ( x ) - sin ( x ) )}

Since the denominators are equal, combine the fractions:

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

= \frac{( 3 cos ( x ) + sin ( x ) ) \cdot 726 - 1835 ( 3 cos ( x ) - sin ( x ) )}{726 ( 3 cos ( x ) - sin ( x ) )}

Expand

(3 cos (x) + sin (x)) \cdot 726 - 1835 (3 cos (x) - sin (x)) : - 11093 cos (x) + 2561 sin (x)

(3 cos (x) + sin (x)) \cdot 726 - 1835 (3 cos (x) - sin (x))

= 726 (3 cos (x) + sin (x)) - 1835 (3 cos (x) - sin (x))

Expand

726 (3 cos (x) + sin (x)) : 7263 cos (x) + 726 sin (x)

726 (3 cos (x) + sin (x))

Apply the distributive law:

a (b + c) = ab + a c

a = 726, b = 3 cos (x), c = sin (x)

= 7263 cos (x) + 726 sin (x)

= 7263 cos (x) + 726 sin (x) - 1835 (3 cos (x) - sin (x))

Expand

- 1835 (3 cos (x) - sin (x)) : - 18353 cos (x) + 1835 sin (x)

- 1835 (3 cos (x) - sin (x))

Apply the distributive law:

a (b - c) = ab - a c

a = - 1835, b = 3 cos (x), c = sin (x)

= - 18353 cos (x) - (- 1835) sin (x)

Apply minus-plus rules

- (- a) = a

= - 18353 cos (x) + 1835 sin (x)

= 7263 cos (x) + 726 sin (x) - 18353 cos (x) + 1835 sin (x)

Simplify

7263 cos (x) + 726 sin (x) - 18353 cos (x) + 1835 sin (x) : - 11093 cos (x) + 2561 sin (x)

7263 cos (x) + 726 sin (x) - 18353 cos (x) + 1835 sin (x)

Add similar elements:

7263 cos (x) - 18353 cos (x) = - 11093 cos (x)

= - 11093 cos (x) + 726 sin (x) + 1835 sin (x)

Add similar elements:

726 sin (x) + 1835 sin (x) = 2561 sin (x)

= - 11093 cos (x) + 2561 sin (x)

= - 11093 cos (x) + 2561 sin (x)

= \frac{- 1109 3 cos ( x ) + 2561 sin ( x )}{726 ( 3 cos ( x ) - sin ( x ) )}

\frac{- 1109 3 cos ( x ) + 2561 sin ( x )}{726 ( 3 cos ( x ) - sin ( x ) )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

- 11093 cos (x) + 2561 sin (x) = 0

Rewrite using trig identities

- 11093 cos (x) + 2561 sin (x) = 0

Divide both sides by

cos (x), cos (x) \neq = 0

\frac{- 1109 3 cos ( x ) + 2561 sin ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

- 11093 + \frac{2561 sin ( x )}{cos ( x )} = 0

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

- 11093 + 2561 tan (x) = 0

- 11093 + 2561 tan (x) = 0

Move

11093

to the right side

- 11093 + 2561 tan (x) = 0

Add

11093

to both sides

- 11093 + 2561 tan (x) + 11093 = 0 + 11093

Simplify

2561 tan (x) = 11093

2561 tan (x) = 11093

Divide both sides by

2561

2561 tan (x) = 11093

Divide both sides by

2561

\frac{2561 tan ( x )}{2561} = \frac{1109 3}{2561}

Simplify

tan (x) = \frac{1109 3}{2561}

tan (x) = \frac{1109 3}{2561}

Apply trig inverse properties

tan (x) = \frac{1109 3}{2561}

General solutions for

tan (x) = \frac{1109 3}{2561}

tan (x) = a \Rightarrow x = arctan (a) + 18 0^{\circ} n

x = arctan (\frac{1109 3}{2561}) + 18 0^{\circ} n

x = arctan (\frac{1109 3}{2561}) + 18 0^{\circ} n

Show solutions in decimal form

x = 0.64352 \dots + 18 0^{\circ} n

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Frequently Asked Questions (FAQ)

What is the general solution for (cos(-30+x))/(cos(30+x))= 1835/726 ?
The general solution for (cos(-30+x))/(cos(30+x))= 1835/726 is x=0.64352…+180n