Let θ be an angle where: 1)θQuadrant III and 2)sin(θ)=1517. What Quadrant does 12θ belong to ? No Calculators !!

1 Answer
Feb 16, 2018

12θ is in Q1.

Explanation:

As θQuadrantIII and sinθ=1517=0.882353

cosθ=1(1517)2=1225289=64289=817=0.470588

Then sin3θ=3sinx4sin3x=3(0.882353)4(0.882353)3

= 2.647059+2.747813=0.100753

and cos3θ=4cos3x3cosx=4(0.470588)33(0.470588)

= 0.416853+1.411764=0.994911

Note that as sin3θ and cos3θ are positive, 3θ is in Q1,

Now sin6θ=2sin3θcos3θ=2×0.100753×0.994911

= 0.200481

and cos6θ=(0.994911)2(0.200481)2=0.949655

and 6θ is in Q2

Continuing this way sina2θ=2sin6θcos6θ=2×0.200481×0.949655

= 0.380776

and cos12θ=(0.949655)2(0.200481)2=0.861652

Hence 12θ is in Q1.