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 !!

Question

1215 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-02-16T05:42:09

$12theta$ is in $Q1$ .

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 $Q1$ .

Let thetatheta be an angle where: "1)" theta in "Quadrant III""1)" theta in "Quadrant III" and "2)" sin( theta ) = - 15/17"2)" sin( theta ) = - 15/17. What Quadrant does 12theta12theta belong to ? No Calculators !!