What is the general solution for cos(sqrt(3)x)+sin(sqrt(3)x)=0 ?

The general solution for cos(sqrt(3)x)+sin(sqrt(3)x)=0 is x=(sqrt(3)pi)/4+(pin)/(sqrt(3))

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Popular >

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

Solution

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

Solution

x = \frac{3 π}{4} + \frac{πn}{3}

Degrees

x = 77.94228 \dots^{\circ} + 103.92304 \dots^{\circ} n

Solution steps

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

Rewrite using trig identities

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

Divide both sides by

cos (3 x), cos (3 x) \neq = 0

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

Simplify

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

Use the basic trigonometric identity:

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

1 + tan (3 x) = 0

1 + tan (3 x) = 0

Move

1

to the right side

1 + tan (3 x) = 0

Subtract

1

from both sides

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

Simplify

tan (3 x) = - 1

tan (3 x) = - 1

General solutions for

tan (3 x) = - 1

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

3 x = \frac{3 π}{4} + πn

3 x = \frac{3 π}{4} + πn

Solve

3 x = \frac{3 π}{4} + πn : x = \frac{3 π}{4} + \frac{πn}{3}

3 x = \frac{3 π}{4} + πn

Divide both sides by

3

3 x = \frac{3 π}{4} + πn

Divide both sides by

3

\frac{3 x}{3} = \frac{\frac{3 π}{4}}{3} + \frac{πn}{3}

Simplify

\frac{3 x}{3} = \frac{\frac{3 π}{4}}{3} + \frac{πn}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Cancel the common factor:

3

= x

Simplify

\frac{\frac{3 π}{4}}{3} + \frac{πn}{3} : \frac{3 π}{4} + \frac{πn}{3}

\frac{\frac{3 π}{4}}{3} + \frac{πn}{3}

\frac{\frac{3 π}{4}}{3} = \frac{3 π}{4}

\frac{\frac{3 π}{4}}{3}

Apply the fraction rule:

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

= \frac{3 π}{4 3}

Apply radical rule:

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

= \frac{3 π}{4 \cdot 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}}

= \frac{3 ^{- \frac{1}{2} + 1} π}{4}

Subtract the numbers:

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

= \frac{3 ^{\frac{1}{2}} π}{4}

Apply radical rule:

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

= \frac{3 π}{4}

= \frac{3 π}{4} + \frac{πn}{3}

x = \frac{3 π}{4} + \frac{πn}{3}

x = \frac{3 π}{4} + \frac{πn}{3}

x = \frac{3 π}{4} + \frac{πn}{3}

x = \frac{3 π}{4} + \frac{πn}{3}

Popular Examples

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

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

sin(θ)=-(sqrt(5))/5

sin (θ) = - \frac{5}{5}

0.9=cos(x)

0.9 = cos (x)

cos(x)= 8/9

cos (x) = \frac{8}{9}

5-6cos(θ)=3sqrt(3)+5-3cos(θ)

5 - 6 cos (θ) = 33 + 5 - 3 cos (θ)

Frequently Asked Questions (FAQ)

What is the general solution for cos(sqrt(3)x)+sin(sqrt(3)x)=0 ?
The general solution for cos(sqrt(3)x)+sin(sqrt(3)x)=0 is x=(sqrt(3)pi)/4+(pin)/(sqrt(3))