How do I solve for the two smallest positive solutions for: sin(2x)cos(6x)-cos(2x)sin(6x) = -0.35 ?

I understand that I am using a sine sum and difference identity (sin(A+B)=sin(A)cos(B)+cos(A)sin(B)) but I have no clue what to do with the negative decimal number at the end of the equation.

This is what I have so far:
sin(2x)cos(6x)cos(2x)sin(6x)=0.35

Then I take sine and simplify.
sin(5x10x)=0.35
sin(5x)=0.35
sin(5x)=0.35

Then I solve for x.
5x=θ
x=15θ

Then I have to solve for θ.
sin(θ)=0.35
sin(θ)=???

Once I figure out how to fine θ, then I'll be able to find the solutions.

2 Answers
Jun 27, 2018

512;3988

Explanation:

This equation comes from the trig identity:
sin (a - b) = sin a.cos b - sin b.cos a.
In this case:
sin (2x - 6x) = sin 2x.cos 6x - sin 6x.cos 2x
sin (2x - 6x) = sin (-4x) = - sin 4x = -0.35
sin 4x = 0.35
Calculator and unit circle give 2 solutions for 4x:
a. 4x=2049+k360 -->
x=512+k90, and
b. 4x=180(20.49)=15951+k360 -->
x=3988+k90
The 2 smallest positive answers are (k = 0):
x=512, and x=3988

Jun 27, 2018

x=5.12 or x=39.88

and in radians x=0.0894 or x=0.696

Explanation:

As sin(2x)cos(6x)cos(2x)sin(6x)=0.35, we have

cos(2x)sin(6x)sin(2x)cos(6x)=0.35

or sin(6x)cos(2x)cos(6x)sin(2x)=0.35 (here we have used commutative property)

As sin(AB)=sinAcosBcosAsinB, we can write the above as

sin(6x2x)=0.35=sin20.49

(We have used scientific calculator to find your θ here.)

and hence either 4x=20.49 i.e. x=5.12

or 4x=18020.49=159.51 i.e. x=39.88

If you need to find in radians sin(0.35757)=0.35

and then 4x=0.35757 or x=0.0894

we can also have 4x=π0.35757=2.78402 and x=0.696

If scientific calculator is not available, one can use tables too.