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

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

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cos(x+pi)=sin(x)

Solution

cos (x + π) = sin (x)

Solution

x = \frac{3 π}{4} + πn

Degrees

x = 13 5^{\circ} + 18 0^{\circ} n

Solution steps

cos (x + π) = sin (x)

Rewrite using trig identities

cos (x + π) = sin (x)

Rewrite using trig identities

cos (x + π)

Use the Angle Sum identity:

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

= cos (x) cos (π) - sin (x) sin (π)

Simplify

cos (x) cos (π) - sin (x) sin (π) : - cos (x)

cos (x) cos (π) - sin (x) sin (π)

cos (x) cos (π) = - cos (x)

cos (x) cos (π)

Simplify

cos (π) : - 1

cos (π)

Use the following trivial identity:

cos (π) = (- 1)

cos (x)

periodicity table with

2 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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= - 1

= (- 1) cos (x)

Refine

= - cos (x)

= - cos (x) - sin (π) sin (x)

sin (x) sin (π) = 0

sin (x) sin (π)

Simplify

sin (π) : 0

sin (π)

Use the following trivial identity:

sin (π) = 0

sin (x)

periodicity table with

2 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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 0

= 0 \cdot sin (x)

Apply rule

0 \cdot a = 0

= 0

= - cos (x) - 0

- cos (x) - 0 = - cos (x)

= - cos (x)

= - cos (x)

- cos (x) = sin (x)

- cos (x) = sin (x)

Subtract

sin (x)

from both sides

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

Rewrite using trig identities

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

Divide both sides by

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

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

Simplify

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

Use the basic trigonometric identity:

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

- 1 - tan (x) = 0

- 1 - tan (x) = 0

Move

1

to the right side

- 1 - tan (x) = 0

Add

1

to both sides

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

Simplify

- tan (x) = 1

- tan (x) = 1

Divide both sides by

- 1

- tan (x) = 1

Divide both sides by

- 1

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

Simplify

tan (x) = - 1

tan (x) = - 1

General solutions for

tan (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}

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

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Frequently Asked Questions (FAQ)

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