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

The general solution for 2sin(x-60)=cos(x-30) is x=1.38067…+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 = 1.38067 \dots + 18 0^{\circ} n

Radians

x = 1.38067 \dots + 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 Difference 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 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)

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

Simplify

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

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

Combine the fractions

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

Apply rule

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

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

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

Convert element to fraction:

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

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

Add similar elements:

- 3 cos (x) - 23 cos (x) = - 33 cos (x)

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

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

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

sin (x) - 33 cos (x) = 0

Rewrite using trig identities

sin (x) - 33 cos (x) = 0

Divide both sides by

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

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

Simplify

\frac{sin ( x )}{cos ( x )} - 33 = 0

Use the basic trigonometric identity:

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

tan (x) - 33 = 0

tan (x) - 33 = 0

Move

33

to the right side

tan (x) - 33 = 0

Add

33

to both sides

tan (x) - 33 + 33 = 0 + 33

Simplify

tan (x) = 33

tan (x) = 33

Apply trig inverse properties

tan (x) = 33

General solutions for

tan (x) = 33

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

x = arctan (33) + 18 0^{\circ} n

x = arctan (33) + 18 0^{\circ} n

Show solutions in decimal form

x = 1.38067 \dots + 18 0^{\circ} n

Popular Examples

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

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=1.38067…+180n