Question

Let `theta` be an angle where: `"1)" theta in "Quadrant III"` and `"2)" sin( theta ) = - 15/17`. What Quadrant does `12theta` belong to ? No Calculators !!

Answer 1

`12theta` is in `Q1`.

Explanation:

As `thetain"Quadrant"III` and `sintheta=-15/17=-0.882353`

`costheta=-sqrt(1-(15/17)^2)=-sqrt(1-225/289)=-sqrt(64/289)=-8/17=-0.470588`

Then `sin3theta=3sinx-4sin^3x=3(-0.882353)-4(-0.882353)^3`

= `-2.647059+2.747813=0.100753`

and `cos3theta=4cos^3x-3cosx=4(-0.470588)^3-3(-0.470588)`

= `-0.416853+1.411764=0.994911`

Note that as `sin3theta` and `cos3theta` are positive, `3theta` is in `Q1`,

Now `sin6theta=2sin3thetacos3theta=2xx0.100753xx0.994911`

= `0.200481`

and `cos6theta=(0.994911)^2-(0.200481)^2=0.949655`

and `6theta` is in `Q2`

Continuing this way `sina2theta=2sin6thetacos6theta=2xx0.200481xx0.949655`

= `0.380776`

and `cos12theta=(0.949655)^2-(0.200481)^2=0.861652`

Hence `12theta` is in `Q1`.