What is the general solution for 5sin(x)=cos(x)-4 ?

The general solution for 5sin(x)=cos(x)-4 is x=-2.04236…+2pin,x=2pi-0.70443…+2pin

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5sin(x)=cos(x)-4

5 sin (x) = cos (x) - 4

x = - 2.04236 \dots + 2 πn, x = 2 π - 0.70443 \dots + 2 πn

Solution steps

5 sin (x) = cos (x) - 4

Square both sides

(5 sin (x))^{2} = (cos (x) - 4)^{2}

Subtract

(cos (x) - 4)^{2}

from both sides

25 sin^{2} (x) - cos^{2} (x) + 8 cos (x) - 16 = 0

Rewrite using trig identities

9 - 26 cos^{2} (x) + 8 cos (x) = 0

Solve by substitution

cos (x) = - \frac{- 4 + 5 10}{26}, cos (x) = \frac{4 + 5 10}{26}

cos (x) = - \frac{- 4 + 5 10}{26} : x = arccos (- \frac{- 4 + 5 10}{26}) + 2 πn, x = - arccos (- \frac{- 4 + 5 10}{26}) + 2 πn

cos (x) = \frac{4 + 5 10}{26} : x = arccos (\frac{4 + 5 10}{26}) + 2 πn, x = 2 π - arccos (\frac{4 + 5 10}{26}) + 2 πn

Combine all the solutions

x = arccos (- \frac{- 4 + 5 10}{26}) + 2 πn, x = - arccos (- \frac{- 4 + 5 10}{26}) + 2 πn, x = arccos (\frac{4 + 5 10}{26}) + 2 πn, x = 2 π - arccos (\frac{4 + 5 10}{26}) + 2 πn

Verify solutions by plugging them into the original equation

x = - arccos (- \frac{- 4 + 5 10}{26}) + 2 πn, x = 2 π - arccos (\frac{4 + 5 10}{26}) + 2 πn

Show solutions in decimal form

x = - 2.04236 \dots + 2 πn, x = 2 π - 0.70443 \dots + 2 πn

tan (x) - 3 = 0

x, 2 sin^{2} (x) - sin (x) - 1 = 0

- 21 tan (x) + 33 = - 3 (3 + tan (x))

tan (y) = - 1

2 π + 6 arccos (2 - x) = 0

What is the general solution for 5sin(x)=cos(x)-4 ?
The general solution for 5sin(x)=cos(x)-4 is x=-2.04236…+2pin,x=2pi-0.70443…+2pin