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

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

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Popular Trigonometry >

cos(x-30)=2sin(x)

Solution

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

Solution

x = 3 0^{\circ} + 18 0^{\circ} n

Radians

x = \frac{π}{6} + πn

Solution steps

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

Rewrite using trig identities

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

Rewrite using trig identities

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

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

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

Subtract

2 sin (x)

from both sides

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

Simplify

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

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

Multiply

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

Multiply

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

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

Multiply fractions:

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

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

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

Apply rule

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

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

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

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

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

Rewrite using trig identities

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

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

3 - 3 tan (x) = 0

3 - 3 tan (x) = 0

Move

3

to the right side

3 - 3 tan (x) = 0

Subtract

3

from both sides

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

Simplify

- 3 tan (x) = - 3

- 3 tan (x) = - 3

Divide both sides by

- 3

- 3 tan (x) = - 3

Divide both sides by

- 3

\frac{- 3 tan ( x )}{- 3} = \frac{- 3}{- 3}

Simplify

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

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

General solutions for

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

tan (x)

periodicity table with

18 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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = 3 0^{\circ} + 18 0^{\circ} n

x = 3 0^{\circ} + 18 0^{\circ} n

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