What is the general solution for 7sin(x)=cos(x)-4 ?

The general solution for 7sin(x)=cos(x)-4 is x=-2.39843…+2pin,x=2pi-0.45936…+2pin

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7sin(x)=cos(x)-4

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

x = - 2.39843 \dots + 2 πn, x = 2 π - 0.45936 \dots + 2 πn

Solution steps

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

Square both sides

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

Subtract

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

from both sides

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

Rewrite using trig identities

33 - 50 cos^{2} (x) + 8 cos (x) = 0

Solve by substitution

cos (x) = - \frac{- 4 + 7 34}{50}, cos (x) = \frac{4 + 7 34}{50}

cos (x) = - \frac{- 4 + 7 34}{50} : x = arccos (- \frac{- 4 + 7 34}{50}) + 2 πn, x = - arccos (- \frac{- 4 + 7 34}{50}) + 2 πn

cos (x) = \frac{4 + 7 34}{50} : x = arccos (\frac{4 + 7 34}{50}) + 2 πn, x = 2 π - arccos (\frac{4 + 7 34}{50}) + 2 πn

Combine all the solutions

x = arccos (- \frac{- 4 + 7 34}{50}) + 2 πn, x = - arccos (- \frac{- 4 + 7 34}{50}) + 2 πn, x = arccos (\frac{4 + 7 34}{50}) + 2 πn, x = 2 π - arccos (\frac{4 + 7 34}{50}) + 2 πn

Verify solutions by plugging them into the original equation

x = - arccos (- \frac{- 4 + 7 34}{50}) + 2 πn, x = 2 π - arccos (\frac{4 + 7 34}{50}) + 2 πn

Show solutions in decimal form

x = - 2.39843 \dots + 2 πn, x = 2 π - 0.45936 \dots + 2 πn

5 sin^{2} (θ) - 11 sin (θ) + 2 = 0

tan^{2} (θ) + 3 sec (θ) = - 3

cos^{2} (x) - sin^{2} (x) = 1 - sin^{2} (x)

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

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

What is the general solution for 7sin(x)=cos(x)-4 ?
The general solution for 7sin(x)=cos(x)-4 is x=-2.39843…+2pin,x=2pi-0.45936…+2pin