What is the general solution for cos(x)=sin(x-pi/2) ?

The general solution for cos(x)=sin(x-pi/2) is x= pi/2+2pin,x=(3pi)/2+2pin

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cos(x)=sin(x-pi/2)

Solution

cos (x) = sin (x - \frac{π}{2})

Solution

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

cos (x) = sin (x - \frac{π}{2})

Rewrite using trig identities

cos (x) = sin (x - \frac{π}{2})

Rewrite using trig identities

sin (x - \frac{π}{2})

Use the Angle Difference identity:

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

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

Simplify

sin (x) cos (\frac{π}{2}) - cos (x) sin (\frac{π}{2}) : - cos (x)

sin (x) cos (\frac{π}{2}) - cos (x) sin (\frac{π}{2})

sin (x) cos (\frac{π}{2}) = 0

sin (x) cos (\frac{π}{2})

Simplify

cos (\frac{π}{2}) : 0

cos (\frac{π}{2})

Use the following trivial identity:

cos (\frac{π}{2}) = 0

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}

= 0

= 0 \cdot sin (x)

Apply rule

0 \cdot a = 0

= 0

cos (x) sin (\frac{π}{2}) = cos (x)

cos (x) sin (\frac{π}{2})

Simplify

sin (\frac{π}{2}) : 1

sin (\frac{π}{2})

Use the following trivial identity:

sin (\frac{π}{2}) = 1

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}

= 1

= 1 \cdot cos (x)

Multiply:

cos (x) \cdot 1 = cos (x)

= cos (x)

= 0 - cos (x)

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

= - cos (x)

= - cos (x)

cos (x) = - cos (x)

cos (x) = - cos (x)

Subtract

- cos (x)

from both sides

2 cos (x) = 0

Divide both sides by

2

2 cos (x) = 0

Divide both sides by

2

\frac{2 cos ( x )}{2} = \frac{0}{2}

Simplify

cos (x) = 0

cos (x) = 0

General solutions for

cos (x) = 0

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}

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

Popular Examples

5cos(x)+2=0,0<= x<= 2pi

5 cos (x) + 2 = 0, 0 \leq x \leq 2 π

2sin(x)cos(x)-sin(2x)cos(2x)=0

2 sin (x) cos (x) - sin (2 x) cos (2 x) = 0

cos(x)= 21/29

cos (x) = \frac{21}{29}

4/(sin(104.48))= 2/(sin(a))

\frac{4}{sin ( 104.4 8 ^{\circ} )} = \frac{2}{sin ( a )}

sin(x+(3pi)/2)-cos(x+pi/2)=1

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

Frequently Asked Questions (FAQ)

What is the general solution for cos(x)=sin(x-pi/2) ?
The general solution for cos(x)=sin(x-pi/2) is x= pi/2+2pin,x=(3pi)/2+2pin