What is the general solution for cos(2θ)-sin(2θ)=0 ?

The general solution for cos(2θ)-sin(2θ)=0 is θ= pi/8+(pin)/2

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Popular Trigonometry >

cos(2θ)-sin(2θ)=0

Solution

cos (2 θ) - sin (2 θ) = 0

Solution

θ = \frac{π}{8} + \frac{πn}{2}

Degrees

θ = 22. 5^{\circ} + 9 0^{\circ} n

Solution steps

cos (2 θ) - sin (2 θ) = 0

Rewrite using trig identities

cos (2 θ) - sin (2 θ) = 0

Divide both sides by

cos (2 θ), cos (2 θ) \neq = 0

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

Simplify

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

Use the basic trigonometric identity:

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

1 - tan (2 θ) = 0

1 - tan (2 θ) = 0

Move

1

to the right side

1 - tan (2 θ) = 0

Subtract

1

from both sides

1 - tan (2 θ) - 1 = 0 - 1

Simplify

- tan (2 θ) = - 1

- tan (2 θ) = - 1

Divide both sides by

- 1

- tan (2 θ) = - 1

Divide both sides by

- 1

\frac{- tan ( 2 θ )}{- 1} = \frac{- 1}{- 1}

Simplify

tan (2 θ) = 1

tan (2 θ) = 1

General solutions for

tan (2 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}

2 θ = \frac{π}{4} + πn

2 θ = \frac{π}{4} + πn

Solve

2 θ = \frac{π}{4} + πn : θ = \frac{π}{8} + \frac{πn}{2}

2 θ = \frac{π}{4} + πn

Divide both sides by

2

2 θ = \frac{π}{4} + πn

Divide both sides by

2

\frac{2 θ}{2} = \frac{\frac{π}{4}}{2} + \frac{πn}{2}

Simplify

\frac{2 θ}{2} = \frac{\frac{π}{4}}{2} + \frac{πn}{2}

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

Simplify

\frac{\frac{π}{4}}{2} + \frac{πn}{2} : \frac{π}{8} + \frac{πn}{2}

\frac{\frac{π}{4}}{2} + \frac{πn}{2}

\frac{\frac{π}{4}}{2} = \frac{π}{8}

\frac{\frac{π}{4}}{2}

Apply the fraction rule:

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

= \frac{π}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{π}{8}

= \frac{π}{8} + \frac{πn}{2}

θ = \frac{π}{8} + \frac{πn}{2}

θ = \frac{π}{8} + \frac{πn}{2}

θ = \frac{π}{8} + \frac{πn}{2}

θ = \frac{π}{8} + \frac{πn}{2}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for cos(2θ)-sin(2θ)=0 ?
The general solution for cos(2θ)-sin(2θ)=0 is θ= pi/8+(pin)/2