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

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

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sin(θ-pi/2)=cos(θ)

Solution

sin (θ - \frac{π}{2}) = cos (θ)

Solution

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

Degrees

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

Solution steps

sin (θ - \frac{π}{2}) = cos (θ)

Rewrite using trig identities

sin (θ - \frac{π}{2}) = cos (θ)

Rewrite using trig identities

sin (θ - \frac{π}{2})

Use the Angle Difference identity:

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

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

Simplify

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

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

sin (θ) cos (\frac{π}{2}) = 0

sin (θ) 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 (θ)

Apply rule

0 \cdot a = 0

= 0

cos (θ) sin (\frac{π}{2}) = cos (θ)

cos (θ) 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 (θ)

Multiply:

cos (θ) \cdot 1 = cos (θ)

= cos (θ)

= 0 - cos (θ)

0 - cos (θ) = - cos (θ)

= - cos (θ)

= - cos (θ)

- cos (θ) = cos (θ)

- cos (θ) = cos (θ)

Subtract

cos (θ)

from both sides

- 2 cos (θ) = 0

Divide both sides by

- 2

- 2 cos (θ) = 0

Divide both sides by

- 2

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

Simplify

cos (θ) = 0

cos (θ) = 0

General solutions for

cos (θ) = 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}

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

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

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Frequently Asked Questions (FAQ)

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