What is the general solution for sin(30-x)+cos(x)=0 ?

The general solution for sin(30-x)+cos(x)=0 is x=60+180n

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

sin(30-x)+cos(x)=0

Solution

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

Solution

x = 6 0^{\circ} + 18 0^{\circ} n

Radians

x = \frac{π}{3} + πn

Solution steps

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

Rewrite using trig identities

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

Rewrite using trig identities

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

Use the Angle Difference identity:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

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

Simplify

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

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

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{1}{2} cos (x) - \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) + cos (x) = 0

Simplify

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

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

Add similar elements:

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

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

Factor out common term

cos (x)

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

\frac{1}{2} + 1 = \frac{3}{2}

\frac{1}{2} + 1

Convert element to fraction:

1 = \frac{1 \cdot 2}{2}

= \frac{1}{2} + \frac{1 \cdot 2}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{1 + 1 \cdot 2}{2}

1 + 1 \cdot 2 = 3

1 + 1 \cdot 2

Multiply the numbers:

1 \cdot 2 = 2

= 1 + 2

Add the numbers:

1 + 2 = 3

= 3

= \frac{3}{2}

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

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

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

\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

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

Simplify

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

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

Apply the fraction rule:

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

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

Cancel the common factor:

3

= tan (x)

Simplify

\frac{- 3}{- 3} : 3

\frac{- 3}{- 3}

Apply the fraction rule:

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

= \frac{3}{3}

Apply radical rule:

3 = 3^{\frac{1}{2}}

= \frac{3}{3 ^{\frac{1}{2}}}

Apply exponent rule:

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

\frac{3 ^{1}}{3 ^{\frac{1}{2}}} = 3^{1 - \frac{1}{2}}

= 3^{1 - \frac{1}{2}}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= 3^{\frac{1}{2}}

Apply radical rule:

3^{\frac{1}{2}} = 3

= 3

tan (x) = 3

tan (x) = 3

tan (x) = 3

General solutions for

tan (x) = 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 = 6 0^{\circ} + 18 0^{\circ} n

x = 6 0^{\circ} + 18 0^{\circ} n

Popular Examples

tan(bx)=5

tan (b x) = 5

sin(t+pi/4)= 1/2 ,pi<= t<= (7pi)/2

sin (t + \frac{π}{4}) = \frac{1}{2}, π \leq t \leq \frac{7 π}{2}

tan(x)=3007

tan (x) = 3007

2sin(θ)-3=0

2 sin (θ) - 3 = 0

2cos^2(t)+7cos(t)+5=0

2 cos^{2} (t) + 7 cos (t) + 5 = 0

Frequently Asked Questions (FAQ)

What is the general solution for sin(30-x)+cos(x)=0 ?
The general solution for sin(30-x)+cos(x)=0 is x=60+180n