Firstly, express the LHS in the form R sin(3θα).

(Note the negative sign and the `3θ`! We have to increase the domain by 3 times.)

`7\ sin 3theta-6\ cos 3theta` `R\ sin(3theta-alpha)`

`R=sqrt(7^2+6^2)` `=sqrt(49+36)=sqrt85`

`alpha=arctan(6/7)=40.60^@`

So

`7\ sin 3theta-6\ cos 3theta` `sqrt85sin(3theta-40.60^@)`

Now, solving the original equation gives:

`sqrt85sin(3theta-40.60^@)` `=3.8`

` sin(3theta-40.60^@)` `=3.8/sqrt85` `=0.412`

Since we have 3θ , we must use the domain: `0^@ ≤ 3θ < 1080^@`.

Sine is positive in Quadrants I,II and V, VI and IX and X.

So, from our calculator, we get the following for `(3θ − 40.60^@)`:

`24.33^@, 155.67^@, 384.33^@,` `515.67^@, 744.33^@ and 875.67^@`

So `3θ` will be:

`64.93^@, 196.27^@, 424.93^@,` `556.27^@, 784.93^@, 916.27^@`.

So the solutions for `θ` are:

`21.6^@, 65.4^@, 141.6^@,` `185.4^@, 261.6^@, 305.4^@`.

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