What is the general solution for 4sin(x)-3cos(x)=5 ?

The general solution for 4sin(x)-3cos(x)=5 is x=2.21429…+2pin

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4sin(x)-3cos(x)=5

4 sin (x) - 3 cos (x) = 5

x = 2.21429 \dots + 2 πn

Solution steps

4 sin (x) - 3 cos (x) = 5

Add

3 cos (x)

to both sides

4 sin (x) = 5 + 3 cos (x)

Square both sides

(4 sin (x))^{2} = (5 + 3 cos (x))^{2}

Subtract

(5 + 3 cos (x))^{2}

from both sides

16 sin^{2} (x) - 25 - 30 cos (x) - 9 cos^{2} (x) = 0

Rewrite using trig identities

- 9 - 25 cos^{2} (x) - 30 cos (x) = 0

Solve by substitution

cos (x) = - \frac{3}{5}

cos (x) = - \frac{3}{5} : x = arccos (- \frac{3}{5}) + 2 πn, x = - arccos (- \frac{3}{5}) + 2 πn

Combine all the solutions

x = arccos (- \frac{3}{5}) + 2 πn, x = - arccos (- \frac{3}{5}) + 2 πn

Verify solutions by plugging them into the original equation

x = arccos (- \frac{3}{5}) + 2 πn

Show solutions in decimal form

x = 2.21429 \dots + 2 πn

x, 3 sin^{2} (x) = cos^{2} (x)

cos (4 0^{\circ})

2 sin (2 x) = sin (x)

sin (\frac{11 π}{4})

arccos (\frac{4}{5})

What is the general solution for 4sin(x)-3cos(x)=5 ?
The general solution for 4sin(x)-3cos(x)=5 is x=2.21429…+2pin