Question

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

Answer 1

`\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&=1\\\\\left((2)/(7)\right)^2+cos^2\theta&=1\\\\(4)/(49)+cos^2\theta&=1\\\\cos^2\theta&=1-(4)/(49)\\\\cos^2\theta&=(45)/(49)\\\\cos\theta&=\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&=2\cos^2\theta -1\\\\&=2 \left(-\sqrt{(45)/(49)}\right)^2-1\\\\&=2 \left((45)/(49)\right)-1\\\\&=(90)/(49)-1\\\\&=(90)/(49)-(49)/(49)\\\\&=(90-49)/(49)\\\\&=(41)/(49)\\\\\end{aligned}`

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