Question #14b3d

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Question #14b3d

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Answer 1 · 2016-12-22T15:16:30

$cot (2 θ) + 5 cos θ = 2$ $cot(2theta)+5costheta=2$

$\frac{cos (2 θ)}{sin (2 θ)} + 5 cos θ = 2$ $cos(2theta)/sin(2theta)+5costheta=2$

$\frac{{cos}^{2} θ - {sin}^{2} θ}{2 sin θ cos θ} + 5 cos θ = 2$ $frac{cos^2theta-sin^2theta}{2sinthetacostheta}+5costheta=2$

$\frac{{cos}^{2} θ - {sin}^{2} θ + (5 cos θ) \cdot (2 sin θ cos θ)}{2 sin θ cos θ} = 2$ $frac{cos^2theta-sin^2theta+(5costheta)*(2sinthetacostheta)}{2sinthetacostheta}=2$

$\frac{{cos}^{2} θ - {sin}^{2} θ + 10 sin θ {cos}^{2} θ}{2 sin θ cos θ} = 2$ $frac{cos^2theta-sin^2theta+10sinthetacos^2theta}{2sinthetacostheta}=2$

$\frac{1 - {sin}^{2} θ - {sin}^{2} θ + 10 sin θ (1 - {sin}^{2} θ)}{2 sin θ cos θ} = 2$ $frac{1-sin^2theta-sin^2theta+10sintheta(1-sin^2theta)}{2sinthetacostheta}=2$

$\frac{1 - 2 {sin}^{2} θ + 10 sin θ - 10 {sin}^{3} θ}{2 sin θ cos θ} = 2$ $frac{1-2sin^2theta+10sintheta-10sin^3theta}{2sinthetacostheta}=2$

$- 10 {sin}^{3} θ - 2 {sin}^{2} θ + 10 sin θ + 1 = 4 sin θ cos θ$ $-10sin^3theta-2sin^2theta+10sintheta+1=4sinthetacostheta$

Using a graphing calculator, I graphed each side of the equation to find the points of intersection.
$θ \approx 1.036 + 2 π n$ $theta approx 1.036 +2pin$
$θ \approx 1.74 + 2 π n$ $thetaapprox 1.74+2pin$
$θ \approx 3.213 + 2 π n$ $thetaapprox 3.213+2pin$

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Question #14b3d

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