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If 90°<0<180° and sin0=2/7, find cos 20.

If 90°<0<180° and sin0=2/7, find cos 20.-example-1
User Fitz
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1 Answer

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Answer:


\textsf{A)} \quad \cos 2 \theta=(41)/(49)

Explanation:

To find the value of cos 2θ given sin θ = 2/7 where 90° < θ < 180°, first use the trigonometric identity sin²θ + cos²θ = 1 to find cos θ:


\begin{aligned}\sin^2\theta+\cos^2\theta&amp;=1\\\\\left((2)/(7)\right)^2+cos^2\theta&amp;=1\\\\(4)/(49)+cos^2\theta&amp;=1\\\\cos^2\theta&amp;=1-(4)/(49)\\\\cos^2\theta&amp;=(45)/(49)\\\\cos\theta&amp;=\pm\sqrt{(45)/(49)}\end{aligned}

Since 90° < θ < 180°, the cosine of θ is in quadrant II of the unit circle, and so cos θ is negative. Therefore:


\boxed{\cos\theta=-\sqrt{(45)/(49)}}

Now we can use the cosine double angle identity to calculate cos 2θ.


\boxed{\begin{minipage}{6.5 cm}\underline{Cosine Double Angle Identity}\\\\$\cos (A \pm B)=\cos A \cos B \mp \sin A \sin B$\\\\$\cos (2 \theta)=\cos^2 \theta - \sin^2 \theta$\\\\$\cos (2 \theta)=2 \cos^2 \theta - 1$\\\\$\cos (2 \theta)=1 - 2 \sin^2 \theta$\\\end{minipage}}

Substitute the value of cos θ:


\begin{aligned}\cos 2\theta&amp;=2\cos^2\theta -1\\\\&amp;=2 \left(-\sqrt{(45)/(49)}\right)^2-1\\\\&amp;=2 \left((45)/(49)\right)-1\\\\&amp;=(90)/(49)-1\\\\&amp;=(90)/(49)-(49)/(49)\\\\&amp;=(90-49)/(49)\\\\&amp;=(41)/(49)\\\\\end{aligned}

Therefore, when 90° < θ < 180° and sin θ = 2/7, the value of cos 2θ is 41/49.

User Fredrik Hedblad
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