What is the general solution for cos(x+30)=2cos(x) ?

The general solution for cos(x+30)=2cos(x) is x=-1.15551…+180n

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cos(x+30)=2cos(x)

Solution

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

Solution

x = - 1.15551 \dots + 18 0^{\circ} n

Radians

x = - 1.15551 \dots + πn

Solution steps

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

Rewrite using trig identities

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

Rewrite using trig identities

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

Use the Angle Sum 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{3}{2} cos (x) - \frac{1}{2} sin (x) = 2 cos (x)

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

Subtract

2 cos (x)

from both sides

- \frac{1}{2} sin (x) + \frac{3 - 4}{2} cos (x) = 0

Simplify

- \frac{1}{2} sin (x) + \frac{3 - 4}{2} cos (x) : \frac{- sin ( x ) + ( 3 - 4 ) cos ( x )}{2}

- \frac{1}{2} sin (x) + \frac{3 - 4}{2} cos (x)

\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{3 - 4}{2} cos (x) = \frac{( 3 - 4 ) cos ( x )}{2}

\frac{3 - 4}{2} cos (x)

Multiply fractions:

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

= \frac{( 3 - 4 ) cos ( x )}{2}

= - \frac{sin ( x )}{2} + \frac{( 3 - 4 ) cos ( x )}{2}

Apply rule

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

= \frac{- sin ( x ) + ( 3 - 4 ) cos ( x )}{2}

\frac{- sin ( x ) + ( 3 - 4 ) cos ( x )}{2} = 0

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

- sin (x) + (3 - 4) cos (x) = 0

Rewrite using trig identities

- sin (x) + (3 - 4) cos (x) = 0

Divide both sides by

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

\frac{- sin ( x ) + ( 3 - 4 ) cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

- \frac{sin ( x )}{cos ( x )} + 3 - 4 = 0

Use the basic trigonometric identity:

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

- tan (x) + 3 - 4 = 0

- tan (x) + 3 - 4 = 0

Move

3

to the right side

- tan (x) + 3 - 4 = 0

Subtract

3

from both sides

- tan (x) + 3 - 4 - 3 = 0 - 3

Simplify

- tan (x) - 4 = - 3

- tan (x) - 4 = - 3

Move

4

to the right side

- tan (x) - 4 = - 3

Add

4

to both sides

- tan (x) - 4 + 4 = - 3 + 4

Simplify

- tan (x) = - 3 + 4

- tan (x) = - 3 + 4

Divide both sides by

- 1

- tan (x) = - 3 + 4

Divide both sides by

- 1

\frac{- tan ( x )}{- 1} = - \frac{3}{- 1} + \frac{4}{- 1}

Simplify

\frac{- tan ( x )}{- 1} = - \frac{3}{- 1} + \frac{4}{- 1}

Simplify

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

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

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= tan (x)

Simplify

- \frac{3}{- 1} + \frac{4}{- 1} : 3 - 4

- \frac{3}{- 1} + \frac{4}{- 1}

Apply rule

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

= \frac{- 3 + 4}{- 1}

Apply the fraction rule:

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

= - \frac{- 3 + 4}{1}

Apply rule

\frac{a}{1} = a

= - (4 - 3)

Distribute parentheses

= - (- 3) - (4)

Apply minus-plus rules

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

= 3 - 4

tan (x) = 3 - 4

tan (x) = 3 - 4

tan (x) = 3 - 4

Apply trig inverse properties

tan (x) = 3 - 4

General solutions for

tan (x) = 3 - 4

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

x = arctan (3 - 4) + 18 0^{\circ} n

x = arctan (3 - 4) + 18 0^{\circ} n

Show solutions in decimal form

x = - 1.15551 \dots + 18 0^{\circ} n

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Frequently Asked Questions (FAQ)

What is the general solution for cos(x+30)=2cos(x) ?
The general solution for cos(x+30)=2cos(x) is x=-1.15551…+180n