What is the general solution for 7tan(x)=8sin(x) ?

The general solution for 7tan(x)=8sin(x) is x=2pin,x=pi+2pin,x=0.50536…+2pin,x=2pi-0.50536…+2pin

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7tan(x)=8sin(x)

7 tan (x) = 8 sin (x)

x = 2 πn, x = π + 2 πn, x = 0.50536 \dots + 2 πn, x = 2 π - 0.50536 \dots + 2 πn

Solution steps

7 tan (x) = 8 sin (x)

Subtract

8 sin (x)

from both sides

7 tan (x) - 8 sin (x) = 0

Express with sin, cos

\frac{7 sin ( x ) - 8 cos ( x ) sin ( x )}{cos ( x )} = 0

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

7 sin (x) - 8 cos (x) sin (x) = 0

Factor

7 sin (x) - 8 cos (x) sin (x) : - sin (x) (8 cos (x) - 7)

- sin (x) (8 cos (x) - 7) = 0

Solving each part separately

sin (x) = 0 or 8 cos (x) - 7 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

8 cos (x) - 7 = 0 : x = arccos (\frac{7}{8}) + 2 πn, x = 2 π - arccos (\frac{7}{8}) + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arccos (\frac{7}{8}) + 2 πn, x = 2 π - arccos (\frac{7}{8}) + 2 πn

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = 0.50536 \dots + 2 πn, x = 2 π - 0.50536 \dots + 2 πn

2 cos (3 x) - 2 = 0

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

4 + 4 sin (x) = 8 cos (x)

- 3 tan^{2} (θ) - 4 tan (θ) = - 4

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

What is the general solution for 7tan(x)=8sin(x) ?
The general solution for 7tan(x)=8sin(x) is x=2pin,x=pi+2pin,x=0.50536…+2pin,x=2pi-0.50536…+2pin