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)

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

x = - 0.38050 \dots + πn

Solution steps

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

Rewrite using trig identities

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}

\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 tan (x) + 2 = 0

Move

2

to the right side

5 tan (x) = - 2

Divide both sides by

5

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

Apply trig inverse properties

x = arctan (- \frac{2}{5}) + πn

Show solutions in decimal form

x = - 0.38050 \dots + πn

cos (x + π) - sin (x - π) = 0

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

sin (2 x) = \frac{( 6 m - 5 )}{5}

4 cos^{2} (x) + 6 = 7

sec (θ) - 1 = (2 - 1) tan (θ)

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