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use an Addition or Subtraction Formula to simplify the equation.

sin(θ) cos(3θ) + cos(θ) sin(3θ) = 0




Find all solutions in the interval [0, 2π). (Enter your answers as a comma-separated list.)

Verified Answer :

Sin (A+B)=sin A cos B+cos A cos B
In this equation:
A=Ф
B=3Ф

Therefore:
sin (Ф) cos(3Ф)+cos (Ф) sin (3Ф)=sin (Ф+3Ф)=sin (4Ф).

Then, we have:
sin (4Ф)=0
4Ф=sin⁻¹ 0=kπ      (k∈Z)    Z=...-2,-1,0,1,2,...
4Ф=kπ
Ф=kπ/4

answer: Ф=Kπ/4    (k∈Z)     Z=...-2,-1,0,1,2...

We can check it out our answer is right.
For example:
if  Z=1; then Ф=π/4
sin(π/4)+cos(3π/4)+cos(π/4)sin(3π/4)=(√2 /2)(-√2/2)+(√2/2)(√2/2)=
=-1/2+1/2=0

The solution in this interval θ=kπ/4

What is Trigonometry?

Trigonometry is the branch of mathematics that deals with the relationship between ratios of the sides of a right-angled triangle with its angles. The ratios used to study this relationship are called trigonometric ratios, namely, sine, cosine, tangent, cotangent, secant, cosecant.

Trigonometry is one of the most important branches in mathematics. The word trigonometry is formed by clubbing words 'Trigonon' and 'Metron' which means triangle and measure respectively. It is the study of the relation between the sides and angles of a right-angled triangle. It thus helps in finding the measure of unknown dimensions of a right-angled triangle using formulas and identities based on this relationship.

As:

sin (θ) cos(3θ)+cos (θ) sin (3θ)

=sin (Ф+3θ)

=sin (4θ).

Then, we have:

sin (4θ)=0

4θ=sin⁻¹ 0=kπ    

4θ=kπ

θ=kπ/4

Learn more about Trigonometry here:

https://brainly.com/question/26719838

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