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

The general solution for sin(x)= 3/5 sin(x+pi/2)+cos(pi-x) is x=-0.38050…+pin

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sin(x)= 3/5 sin(x+pi/2)+cos(pi-x)

Solution

sin (x) = \frac{3}{5} sin (x + \frac{π}{2}) + cos (π - x)

Solution

x = - 0.38050 \dots + πn

Degrees

x = - 21.80140 \dots^{\circ} + 18 0^{\circ} n

Solution steps

sin (x) = \frac{3}{5} sin (x + \frac{π}{2}) + cos (π - x)

Rewrite using trig identities

sin (x) = \frac{3}{5} sin (x + \frac{π}{2}) + cos (π - x)

Rewrite using trig identities

cos (π - x)

Use the Angle Difference identity:

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

= cos (π) cos (x) + sin (π) sin (x)

Simplify

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

cos (π) cos (x) + sin (π) sin (x)

cos (π) cos (x) = - cos (x)

cos (π) cos (x)

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 \cdot cos (x)

Multiply:

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

= - cos (x)

= - cos (x) + sin (π) sin (x)

sin (π) sin (x) = 0

sin (π) sin (x)

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)

Use the Angle Sum 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)

sin (x) = \frac{3}{5} cos (x) - cos (x)

Simplify

\frac{3}{5} cos (x) - cos (x) : - \frac{2}{5} cos (x)

\frac{3}{5} cos (x) - cos (x)

Factor out common term

cos (x)

= cos (x) (\frac{3}{5} - 1)

\frac{3}{5} - 1 = - \frac{2}{5}

\frac{3}{5} - 1

Convert element to fraction:

1 = \frac{1 \cdot 5}{5}

= \frac{3}{5} - \frac{1 \cdot 5}{5}

Since the denominators are equal, combine the fractions:

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

= \frac{3 - 1 \cdot 5}{5}

3 - 1 \cdot 5 = - 2

3 - 1 \cdot 5

Multiply the numbers:

1 \cdot 5 = 5

= 3 - 5

Subtract the numbers:

3 - 5 = - 2

= - 2

= \frac{- 2}{5}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{2}{5}

= - \frac{2}{5} cos (x)

sin (x) = - \frac{2}{5} cos (x)

sin (x) = - \frac{2}{5} cos (x)

Subtract

- \frac{2}{5} cos (x)

from both sides

sin (x) + \frac{2}{5} cos (x) = 0

Simplify

sin (x) + \frac{2}{5} cos (x) : \frac{5 sin ( x ) + 2 cos ( x )}{5}

sin (x) + \frac{2}{5} cos (x)

Multiply

\frac{2}{5} cos (x) : \frac{2 cos ( x )}{5}

\frac{2}{5} cos (x)

Multiply fractions:

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

= \frac{2 cos ( x )}{5}

= sin (x) + \frac{2 cos ( x )}{5}

Convert element to fraction:

sin (x) = \frac{sin ( x ) 5}{5}

= \frac{sin ( x ) \cdot 5}{5} + \frac{2 cos ( x )}{5}

Since the denominators are equal, combine the fractions:

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

= \frac{sin ( x ) \cdot 5 + 2 cos ( x )}{5}

\frac{5 sin ( x ) + 2 cos ( x )}{5} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

5 sin (x) + 2 cos (x) = 0

Rewrite using trig identities

5 sin (x) + 2 cos (x) = 0

Divide both sides by

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

\frac{5 sin ( x ) + 2 cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

\frac{5 sin ( x )}{cos ( x )} + 2 = 0

Use the basic trigonometric identity:

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

5 tan (x) + 2 = 0

5 tan (x) + 2 = 0

Move

2

to the right side

5 tan (x) + 2 = 0

Subtract

2

from both sides

5 tan (x) + 2 - 2 = 0 - 2

Simplify

5 tan (x) = - 2

5 tan (x) = - 2

Divide both sides by

5

5 tan (x) = - 2

Divide both sides by

5

\frac{5 tan ( x )}{5} = \frac{- 2}{5}

Simplify

tan (x) = - \frac{2}{5}

tan (x) = - \frac{2}{5}

Apply trig inverse properties

tan (x) = - \frac{2}{5}

General solutions for

tan (x) = - \frac{2}{5}

tan (x) = - a \Rightarrow x = arctan (- a) + πn

x = arctan (- \frac{2}{5}) + πn

x = arctan (- \frac{2}{5}) + πn

Show solutions in decimal form

x = - 0.38050 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x)= 3/5 sin(x+pi/2)+cos(pi-x) ?
The general solution for sin(x)= 3/5 sin(x+pi/2)+cos(pi-x) is x=-0.38050…+pin