What is the general solution for -sin(x)+3cos(x-pi/2)=1 ?

The general solution for -sin(x)+3cos(x-pi/2)=1 is x= pi/6+2pin,x=(5pi)/6+2pin

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Popular >

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

Solution

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

Solution

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Degrees

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Rewrite using trig identities

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

Rewrite using trig identities

cos (x - \frac{π}{2})

Use the Angle Difference identity:

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

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

Simplify

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

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

cos (x) cos (\frac{π}{2}) = 0

cos (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 cos (x)

Apply rule

0 \cdot a = 0

= 0

sin (x) sin (\frac{π}{2}) = sin (x)

sin (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 sin (x)

Multiply:

sin (x) \cdot 1 = sin (x)

= sin (x)

= 0 + sin (x)

0 + sin (x) = sin (x)

= sin (x)

= sin (x)

- sin (x) + 3 sin (x) = 1

Simplify

- sin (x) + 3 sin (x) : 2 sin (x)

- sin (x) + 3 sin (x)

Add similar elements:

- sin (x) + 3 sin (x) = 2 sin (x)

= 2 sin (x)

2 sin (x) = 1

2 sin (x) = 1

Subtract

1

from both sides

2 sin (x) - 1 = 0

Move

1

to the right side

2 sin (x) - 1 = 0

Add

1

to both sides

2 sin (x) - 1 + 1 = 0 + 1

Simplify

2 sin (x) = 1

2 sin (x) = 1

Divide both sides by

2

2 sin (x) = 1

Divide both sides by

2

\frac{2 sin ( x )}{2} = \frac{1}{2}

Simplify

sin (x) = \frac{1}{2}

sin (x) = \frac{1}{2}

General solutions for

sin (x) = \frac{1}{2}

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}

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Popular Examples

cos(30+u)=0.8177

cos (3 0^{\circ} + u) = 0.8177

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

cos (x - \frac{3 π}{2}) = \frac{1}{2}

1-3cos(2θ)=0

1 - 3 cos (2 θ) = 0

solvefor x,sin(pi/x)=y-1 solve for

x, sin (\frac{π}{x}) = y - 1

cos((3θ)/2)=0

cos (\frac{3 θ}{2}) = 0

Frequently Asked Questions (FAQ)

What is the general solution for -sin(x)+3cos(x-pi/2)=1 ?
The general solution for -sin(x)+3cos(x-pi/2)=1 is x= pi/6+2pin,x=(5pi)/6+2pin