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

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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(5x-10x) = -0.35#

#sin(-5x) = -0.35#

#-sin(5x) = - 0.35#

Then I solve for #x# .

#5x = theta#

#x = 1/5theta#

Then I have to solve for #theta# .

#sin(theta) = -0.35#

#sin(theta) = ???#

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

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:

Then I take sine and simplify.

Then I solve for

Then I have to solve for

Once I figure out how to fine

##### 2 Answers

#### 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.

b.

The 2 smallest positive answers are (k = 0):

and in radians

#### Explanation:

As

or

As

(We have used scientific calculator to find your

and hence either

or

If you need to find in radians

and then

we can also have

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