What is the general solution for 12cos(x)+5sin(x)=3 ?

The general solution for 12cos(x)+5sin(x)=3 is x=-0.94313…+2pin,x=pi-1.40887…+2pin

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12cos(x)+5sin(x)=3

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

x = - 0.94313 \dots + 2 πn, x = π - 1.40887 \dots + 2 πn

Solution steps

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

Subtract

5 sin (x)

from both sides

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

Square both sides

(12 cos (x))^{2} = (3 - 5 sin (x))^{2}

Subtract

(3 - 5 sin (x))^{2}

from both sides

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

Rewrite using trig identities

135 - 169 sin^{2} (x) + 30 sin (x) = 0

Solve by substitution

sin (x) = - \frac{3 ( 16 10 - 5 )}{169}, sin (x) = \frac{3 ( 5 + 16 10 )}{169}

sin (x) = - \frac{3 ( 16 10 - 5 )}{169} : x = arcsin (- \frac{3 ( 16 10 - 5 )}{169}) + 2 πn, x = π + arcsin (\frac{3 ( 16 10 - 5 )}{169}) + 2 πn

sin (x) = \frac{3 ( 5 + 16 10 )}{169} : x = arcsin (\frac{3 ( 5 + 16 10 )}{169}) + 2 πn, x = π - arcsin (\frac{3 ( 5 + 16 10 )}{169}) + 2 πn

Combine all the solutions

x = arcsin (- \frac{3 ( 16 10 - 5 )}{169}) + 2 πn, x = π + arcsin (\frac{3 ( 16 10 - 5 )}{169}) + 2 πn, x = arcsin (\frac{3 ( 5 + 16 10 )}{169}) + 2 πn, x = π - arcsin (\frac{3 ( 5 + 16 10 )}{169}) + 2 πn

Verify solutions by plugging them into the original equation

x = arcsin (- \frac{3 ( 16 10 - 5 )}{169}) + 2 πn, x = π - arcsin (\frac{3 ( 5 + 16 10 )}{169}) + 2 πn

Show solutions in decimal form

x = - 0.94313 \dots + 2 πn, x = π - 1.40887 \dots + 2 πn

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

2 cos (4 θ) = - 1, 0 \leq θ \leq 2 π

1 \cdot sin (4 5^{\circ}) = 1.57 \cdot sin (θ)

\frac{sin ( x )}{2 - 2 sin ^{2} ( x )} = 0

tan (θ) = 3, 0 \leq θ < 2 π

What is the general solution for 12cos(x)+5sin(x)=3 ?
The general solution for 12cos(x)+5sin(x)=3 is x=-0.94313…+2pin,x=pi-1.40887…+2pin