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

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

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Answer 1 · 2018-02-16T05:42:09

$12 θ$ $12theta$ is in $Q 1$ $Q1$ .

Explanation:

As $θ \in Quadrant I I I$ $thetain"Quadrant"III$ and $sin θ = - \frac{15}{17} = - 0.882353$ $sintheta=-15/17=-0.882353$

$cos θ = - \sqrt{1 - {(\frac{15}{17})}^{2}} = - \sqrt{1 - \frac{225}{289}} = - \sqrt{\frac{64}{289}} = - \frac{8}{17} = - 0.470588$ $costheta=-sqrt(1-(15/17)^2)=-sqrt(1-225/289)=-sqrt(64/289)=-8/17=-0.470588$

Then $sin 3 θ = 3 sin x - 4 {sin}^{3} x = 3 (- 0.882353) - 4 {(- 0.882353)}^{3}$ $sin3theta=3sinx-4sin^3x=3(-0.882353)-4(-0.882353)^3$

= $- 2.647059 + 2.747813 = 0.100753$ $-2.647059+2.747813=0.100753$

and $cos 3 θ = 4 {cos}^{3} x - 3 cos x = 4 {(- 0.470588)}^{3} - 3 (- 0.470588)$ $cos3theta=4cos^3x-3cosx=4(-0.470588)^3-3(-0.470588)$

= $- 0.416853 + 1.411764 = 0.994911$ $-0.416853+1.411764=0.994911$

Note that as $sin 3 θ$ $sin3theta$ and $cos 3 θ$ $cos3theta$ are positive, $3 θ$ $3theta$ is in $Q 1$ $Q1$ ,

Now $sin 6 θ = 2 sin 3 θ cos 3 θ = 2 \times 0.100753 \times 0.994911$ $sin6theta=2sin3thetacos3theta=2xx0.100753xx0.994911$

= $0.200481$ $0.200481$

and $cos 6 θ = {(0.994911)}^{2} - {(0.200481)}^{2} = 0.949655$ $cos6theta=(0.994911)^2-(0.200481)^2=0.949655$

and $6 θ$ $6theta$ is in $Q 2$ $Q2$

Continuing this way $sin a 2 θ = 2 sin 6 θ cos 6 θ = 2 \times 0.200481 \times 0.949655$ $sina2theta=2sin6thetacos6theta=2xx0.200481xx0.949655$

= $0.380776$ $0.380776$

and $cos 12 θ = {(0.949655)}^{2} - {(0.200481)}^{2} = 0.861652$ $cos12theta=(0.949655)^2-(0.200481)^2=0.861652$

Hence $12 θ$ $12theta$ is in $Q 1$ $Q1$ .

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

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Let thetaθtheta be an angle where: "1)" theta in "Quadrant III"1)θ∈Quadrant III"1)" theta in "Quadrant III" and "2)" sin( theta ) = - 15/172)sin(θ)=−1517"2)" sin( theta ) = - 15/17. What Quadrant does 12theta12θ12theta belong to ? No Calculators !!

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