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

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

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Popular >

sin(x+30)=cos(x)

Solution

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

Solution

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

Radians

x = \frac{π}{6} + πn

Solution steps

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

Rewrite using trig identities

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

Rewrite using trig identities

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

Use the Angle Sum identity:

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

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

Simplify

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

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

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

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

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

Subtract

cos (x)

from both sides

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

Simplify

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

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

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

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

Multiply fractions:

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

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

Multiply:

1 \cdot cos (x) = cos (x)

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

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

Apply rule

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

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

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

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

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

Rewrite using trig identities

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

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

3 tan (x) - 1 = 0

3 tan (x) - 1 = 0

Move

1

to the right side

3 tan (x) - 1 = 0

Add

1

to both sides

3 tan (x) - 1 + 1 = 0 + 1

Simplify

3 tan (x) = 1

3 tan (x) = 1

Divide both sides by

3

3 tan (x) = 1

Divide both sides by

3

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

Simplify

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

Simplify

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

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

Cancel the common factor:

3

= tan (x)

Simplify

\frac{1}{3} : \frac{3}{3}

\frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{3}

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

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

Popular Examples

sin(pi/6 x)=0

sin (\frac{π}{6} x) = 0

sec(x)-2cos(x)=1

sec (x) - 2 cos (x) = 1

2sin(x)+3=0

2 sin (x) + 3 = 0

4cos(2θ)-10cos(θ)+14=7

4 cos (2 θ) - 10 cos (θ) + 14 = 7

2cos^2(2x)=cos(2x)

2 cos^{2} (2 x) = cos (2 x)

Frequently Asked Questions (FAQ)

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