Question

If cos θ= 12 /13 and θ is located in the Quadrant I, find sin (2 θ ), cos(2 θ ), tan(2 θ )

Answer 1

`\cos 2 \theta = (119)/(169)\\\\\sin2 \theta = (120)/(169)\\\\\tan 2\theta = (120)/(119)`

Explanation:

`\text{Given that,} \cos \theta = (12)/(13)\\ \\\text{Now,}\\\\\cos 2 \theta = 2 \cos^2 \theta - 1\\\\\\~~~~~~~~~=2 \left( (12)/(13) \right)^2 - 1\\\\\\~~~~~~~~~=2 \left( (144)/(169) \right) - 1\\\\\\~~~~~~~~~=(288)/(169)-1\\\\\\~~~~~~~~~=(119)/(169)`

`\sin 2\theta = 2 \sin \theta \cos \theta\\\\\\~~~~~~~~=2√(1 -\cos^2 \theta) \cdot \cos \theta~~~~~~~~~~~;[\text{In quadrant I, all ratios are positive.}]\\\\\\~~~~~~~~=2 \sqrt{1- \left( (12)/(13) \right)^2} \cdot \left((12)/(13) \right)\\\\\\~~~~~~~~=\left( (24)/(13) \right) \sqrt{1- (144)/(169)}\\\\\\~~~~~~~~=(24)/(13)\sqrt{(25)/(169)}\\\\\\~~~~~~~=(24)/(13) * (5)/(13)\\\\\\~~~~~~=(120)/(169)`

`\tan 2 \theta = (\sin 2\theta)/(\cos 2\theta)\\\\\\~~~~~~~~~=\frac{\tfrac{120}{169}}{ \tfrac{119}{169}}\\\\\\~~~~~~~~~=(120)/(169) * (169)/(119)\\\\\\~~~~~~~~~=(120)/(119)`