What is the general solution for 2sin(x+60)=cos(x+30) ?

The general solution for 2sin(x+60)=cos(x+30) is x=150+180n

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2sin(x+60)=cos(x+30)

Solution

2 sin (x + 6 0^{\circ}) = cos (x + 3 0^{\circ})

Solution

x = 15 0^{\circ} + 18 0^{\circ} n

Radians

x = \frac{5 π}{6} + πn

Solution steps

2 sin (x + 6 0^{\circ}) = cos (x + 3 0^{\circ})

Rewrite using trig identities

2 sin (x + 6 0^{\circ}) = cos (x + 3 0^{\circ})

Rewrite using trig identities

sin (x + 6 0^{\circ})

Use the Angle Sum identity:

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

= sin (x) cos (6 0^{\circ}) + cos (x) sin (6 0^{\circ})

Simplify

sin (x) cos (6 0^{\circ}) + cos (x) sin (6 0^{\circ}) : \frac{1}{2} sin (x) + \frac{3}{2} cos (x)

sin (x) cos (6 0^{\circ}) + cos (x) sin (6 0^{\circ})

Simplify

cos (6 0^{\circ}) : \frac{1}{2}

cos (6 0^{\circ})

Use the following trivial identity:

cos (6 0^{\circ}) = \frac{1}{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{1}{2}

= \frac{1}{2} sin (x) + sin (6 0^{\circ}) cos (x)

Simplify

sin (6 0^{\circ}) : \frac{3}{2}

sin (6 0^{\circ})

Use the following trivial identity:

sin (6 0^{\circ}) = \frac{3}{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{3}{2}

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

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

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)

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

Simplify

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

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

Apply the distributive law:

a (b + c) = ab + a c

a = 2, b = \frac{1}{2} sin (x), c = \frac{3}{2} cos (x)

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

Simplify

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

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

= sin (x) \cdot 1

Multiply:

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

= sin (x)

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

= cos (x) 3

= sin (x) + 3 cos (x)

= sin (x) + 3 cos (x)

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

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

Subtract

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

from both sides

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

Simplify

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

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

Combine the fractions

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

Apply rule

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

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Add similar elements:

- 3 cos (x) + 23 cos (x) = 3 cos (x)

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

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

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

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

Rewrite using trig identities

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

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

3 tan (x) + 3 = 0

3 tan (x) + 3 = 0

Move

3

to the right side

3 tan (x) + 3 = 0

Subtract

3

from both sides

3 tan (x) + 3 - 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 = 15 0^{\circ} + 18 0^{\circ} n

x = 15 0^{\circ} + 18 0^{\circ} n

Popular Examples

cot^2(θ)-3csc(θ)=-3

cot^{2} (θ) - 3 csc (θ) = - 3

tan^3(x)= 1/3 tan(x)

tan^{3} (x) = \frac{1}{3} tan (x)

cos^2(x)= 2/3

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

5|tan(x)|=5

5 ∣ tan (x) ∣ = 5

3arccos(x)=2

3 arccos (x) = 2

Frequently Asked Questions (FAQ)

What is the general solution for 2sin(x+60)=cos(x+30) ?
The general solution for 2sin(x+60)=cos(x+30) is x=150+180n