What is the general solution for cos(3θ)=sin(3θ) ?

The general solution for cos(3θ)=sin(3θ) is θ= pi/(12)+(pin)/3

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cos(3θ)=sin(3θ)

cos (3 θ) = sin (3 θ)

θ = \frac{π}{12} + \frac{πn}{3}

Degrees

θ = 1 5^{\circ} + 6 0^{\circ} n

Solution steps

cos (3 θ) = sin (3 θ)

Rewrite using trig identities

cos (3 θ) = sin (3 θ)

Divide both sides by

cos (3 θ), cos (3 θ) \neq = 0

\frac{cos ( 3 θ )}{cos ( 3 θ )} = \frac{sin ( 3 θ )}{cos ( 3 θ )}

Simplify

1 = \frac{sin ( 3 θ )}{cos ( 3 θ )}

Use the basic trigonometric identity:

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

1 = tan (3 θ)

Switch sides

tan (3 θ) = 1

tan (3 θ) = 1

General solutions for

tan (3 θ) = 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 θ = \frac{π}{4} + πn

3 θ = \frac{π}{4} + πn

Solve

3 θ = \frac{π}{4} + πn : θ = \frac{π}{12} + \frac{πn}{3}

3 θ = \frac{π}{4} + πn

Divide both sides by

3

3 θ = \frac{π}{4} + πn

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 θ}{3} : θ

\frac{3 θ}{3}

Divide the numbers:

\frac{3}{3} = 1

= θ

Simplify

\frac{\frac{π}{4}}{3} + \frac{πn}{3} : \frac{π}{12} + \frac{πn}{3}

\frac{\frac{π}{4}}{3} + \frac{πn}{3}

\frac{\frac{π}{4}}{3} = \frac{π}{12}

\frac{\frac{π}{4}}{3}

Apply the fraction rule:

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

= \frac{π}{4 \cdot 3}

Multiply the numbers:

4 \cdot 3 = 12

= \frac{π}{12}

= \frac{π}{12} + \frac{πn}{3}

θ = \frac{π}{12} + \frac{πn}{3}

θ = \frac{π}{12} + \frac{πn}{3}

θ = \frac{π}{12} + \frac{πn}{3}

θ = \frac{π}{12} + \frac{πn}{3}

What is the general solution for cos(3θ)=sin(3θ) ?
The general solution for cos(3θ)=sin(3θ) is θ= pi/(12)+(pin)/3