What is the general solution for 9sin(x)=cos(x)-7 ?

The general solution for 9sin(x)=cos(x)-7 is x=-2.14734…+2pin,x=2pi-0.77293…+2pin

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9sin(x)=cos(x)-7

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

x = - 2.14734 \dots + 2 πn, x = 2 π - 0.77293 \dots + 2 πn

Solution steps

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

Square both sides

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

Subtract

(cos (x) - 7)^{2}

from both sides

81 sin^{2} (x) - cos^{2} (x) + 14 cos (x) - 49 = 0

Rewrite using trig identities

32 + 14 cos (x) - 82 cos^{2} (x) = 0

Solve by substitution

cos (x) = - \frac{- 7 + 9 33}{82}, cos (x) = \frac{7 + 9 33}{82}

cos (x) = - \frac{- 7 + 9 33}{82} : x = arccos (- \frac{- 7 + 9 33}{82}) + 2 πn, x = - arccos (- \frac{- 7 + 9 33}{82}) + 2 πn

cos (x) = \frac{7 + 9 33}{82} : x = arccos (\frac{7 + 9 33}{82}) + 2 πn, x = 2 π - arccos (\frac{7 + 9 33}{82}) + 2 πn

Combine all the solutions

x = arccos (- \frac{- 7 + 9 33}{82}) + 2 πn, x = - arccos (- \frac{- 7 + 9 33}{82}) + 2 πn, x = arccos (\frac{7 + 9 33}{82}) + 2 πn, x = 2 π - arccos (\frac{7 + 9 33}{82}) + 2 πn

Verify solutions by plugging them into the original equation

x = - arccos (- \frac{- 7 + 9 33}{82}) + 2 πn, x = 2 π - arccos (\frac{7 + 9 33}{82}) + 2 πn

Show solutions in decimal form

x = - 2.14734 \dots + 2 πn, x = 2 π - 0.77293 \dots + 2 πn

tan (θ) = \frac{1}{- 3}

sin^{2} (θ) - cos^{2} (θ) = sin (θ)

4 sin (c) + 3 = sin (c) + 1

(tan (x) + 1) (3 tan (x) - 1) = 0

tan^{2} (x) sin (x) - sin (x) = 0

What is the general solution for 9sin(x)=cos(x)-7 ?
The general solution for 9sin(x)=cos(x)-7 is x=-2.14734…+2pin,x=2pi-0.77293…+2pin