What is the general solution for cos(5t)cos(3t)= 1/2+sin(-5t)sin(3t) ?

The general solution for cos(5t)cos(3t)= 1/2+sin(-5t)sin(3t) is t= pi/6+pin,t=(5pi)/6+pin

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cos(5t)cos(3t)= 1/2+sin(-5t)sin(3t)

cos (5 t) cos (3 t) = \frac{1}{2} + sin (- 5 t) sin (3 t)

t = \frac{π}{6} + πn, t = \frac{5 π}{6} + πn

Solution steps

cos (5 t) cos (3 t) = \frac{1}{2} + sin (- 5 t) sin (3 t)

Rewrite using trig identities

cos (5 t) cos (3 t) = \frac{1}{2} - sin (5 t) sin (3 t)

Subtract

\frac{1}{2} - sin (5 t) sin (3 t)

from both sides

cos (5 t) cos (3 t) - \frac{1}{2} + sin (5 t) sin (3 t) = 0

Simplify

cos (5 t) cos (3 t) - \frac{1}{2} + sin (5 t) sin (3 t) : \frac{2 cos ( 5 t ) cos ( 3 t ) - 1 + 2 sin ( 5 t ) sin ( 3 t )}{2}

\frac{2 cos ( 5 t ) cos ( 3 t ) - 1 + 2 sin ( 5 t ) sin ( 3 t )}{2} = 0

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

2 cos (5 t) cos (3 t) - 1 + 2 sin (5 t) sin (3 t) = 0

Rewrite using trig identities

- 1 + 2 cos (2 t) = 0

Move

1

to the right side

2 cos (2 t) = 1

Divide both sides by

2

cos (2 t) = \frac{1}{2}

General solutions for

cos (2 t) = \frac{1}{2}

2 t = \frac{π}{3} + 2 πn, 2 t = \frac{5 π}{3} + 2 πn

Solve

2 t = \frac{π}{3} + 2 πn : t = \frac{π}{6} + πn

Solve

2 t = \frac{5 π}{3} + 2 πn : t = \frac{5 π}{6} + πn

t = \frac{π}{6} + πn, t = \frac{5 π}{6} + πn

sin (3 θ + 7 2^{\circ}) = cos (4 8^{\circ}), 0^{\circ} \leq θ \leq 36 0^{\circ}

cot^{2} (y) + csc (y) - 5 = 0

2 cos (θ) = 1, 0 \leq θ < 2 π

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

3 tan (x) - 1 = 0, 0 \leq x \leq 2 π

What is the general solution for cos(5t)cos(3t)= 1/2+sin(-5t)sin(3t) ?
The general solution for cos(5t)cos(3t)= 1/2+sin(-5t)sin(3t) is t= pi/6+pin,t=(5pi)/6+pin