Question

Solve the trigonometric equation `sin(x+60^@)=2cos(x-45^@)`?

Answer 1

Solution is `21.85^@` and `201.85^@`.

Explanation:

`sin(x+60^@)=2cos(x-45^@)` can be expanded as

`sinxcos60^@+cosxsin60^@=2(cosxcos45^@-sinxsin45^@)`

or `sinx xxsqrt3/2+cosx xx1/2=2(cosx xx1/sqrt2-sinx xx1/sqrt2)`

or `sqrt3sinx+cosx=2sqrt2cosx-2sqrt2sinx`

or `sinx(sqrt3+2sqrt2)=(2sqrt2-1)cosx`

or `sinx/cosx=(2sqrt2-1)/(sqrt3+2sqrt2)=(2xx1.4142-1)/(1.732+2xx1.4142)`

= `1.8284/4.5604=0.401`

i.e. `tanx=0.401=tan21.85^@`

i.e. `x=nxx180^@+21.85^@`, where `n` is an integer.

And in the interval `0^@<=x<=360^@`,

Solution is `21.85^@` and `201.85^@`.