use an addition or subtraction formula to find the solutions of the equations that are in the interval [0,pi). sin 3t cos t + cos 3t sin t = 1/2

Answer 1

The identity for the sine of a sum of angles will help us a lot here:

sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

If we let a = 3t and b = t, we can use this identity backwards to rewrite

sin(3t)cos(t) + cos(3t)sin(t) = sin(3t + t) = sin(4t)

I'll make the substitution x = 4t so we can solve the simplified expression sin(4t) = 1/2 as sin(x) = 1/2. On the interval [0, π) sin(x) = 1/2 has two solutions, x = π/6 and x = 5π/6. Subsituting 4t back in for x gives the expression 4t = π/6 and 4t = 5π/6, which we can solve for t, giving us the solutions

4t = π/6 → t = π/24

4t = 5π/6 → t = 5π/24