Question

Use an addition or subtraction formula to simplify the equation. sin(3θ) cos(θ) â cos(3θ) sin(θ) = 0. Find all solutions in the interval [0, 2].

Answer 1

The solutions in the interval [0, 2] is the θ=0,π/2,π,π3/2.

According to the statement

we have to find that the all solutions of the given statement in the interval [0, 2].

So, For this purpose,

So According to the trigonometry

now lets take the inverse sin of both sides to solve,

sin^-1(sin(2θ))=sin^-1(0)

2θ=sin-1(0)

So you can either plug that into a calculator or just remember (from the unit circle) that the sine is equal to zero at 0 AND at π

2θ=0,π

But we're not done because the sine is periodic which means there will be solutions every trip around the unit circle (2π)

So 2θ=0+N^2π and π+N^2π where N=0,+/-1,+/-2,etc...

But we're looking for values of θ so we have to divide everything by 2

θ=0+Nπ and π/2+Nπ

Now we just have to find the values of N that yield solutions on the interval [0,2π]

N=0 gives 0 and π/2

N=1 gives π and π3/2

So the full solution set is

θ=0,π/2,π,π3/2;

So, The solutions in the interval [0, 2] is the θ=0,π/2,π,π3/2.

Learn more about interval here

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