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Math  /  Trigonometry

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If cos⁡θ=426\cos \theta=\frac{4}{\sqrt{26}} and angle θ\theta is in Quadrant I, what is the exact value of tan⁡2θ\tan 2 \theta in simplest radical form?

Studdy Solution

STEP 1

1. The angle θ\theta is in Quadrant I, where all trigonometric functions are positive.
2. We are given cos⁡θ=426\cos \theta = \frac{4}{\sqrt{26}}.
3. We need to find tan⁡2θ\tan 2\theta using trigonometric identities.

STEP 2

1. Find sin⁡θ\sin \theta using the Pythagorean identity.
2. Use the double angle identity for tangent to find tan⁡2θ\tan 2\theta.

STEP 3

Use the Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1 to find sin⁡θ\sin \theta.
cos⁡θ=426\cos \theta = \frac{4}{\sqrt{26}}
cos⁡2θ=(426)2=1626=813\cos^2 \theta = \left(\frac{4}{\sqrt{26}}\right)^2 = \frac{16}{26} = \frac{8}{13}
sin⁡2θ=1−cos⁡2θ=1−813=513\sin^2 \theta = 1 - \cos^2 \theta = 1 - \frac{8}{13} = \frac{5}{13}
Since θ\theta is in Quadrant I, sin⁡θ\sin \theta is positive:
sin⁡θ=513\sin \theta = \sqrt{\frac{5}{13}}

STEP 4

Use the double angle identity for tangent:
tan⁡2θ=2tan⁡θ1−tan⁡2θ\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta}
First, find tan⁡θ\tan \theta:
tan⁡θ=sin⁡θcos⁡θ=513426\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\sqrt{\frac{5}{13}}}{\frac{4}{\sqrt{26}}}
Simplify tan⁡θ\tan \theta:
tan⁡θ=5/13⋅264=5⋅24=104\tan \theta = \frac{\sqrt{5/13} \cdot \sqrt{26}}{4} = \frac{\sqrt{5} \cdot \sqrt{2}}{4} = \frac{\sqrt{10}}{4}
Now, substitute tan⁡θ\tan \theta into the double angle formula:
tan⁡2θ=2⋅1041−(104)2\tan 2\theta = \frac{2 \cdot \frac{\sqrt{10}}{4}}{1 - \left(\frac{\sqrt{10}}{4}\right)^2}
Simplify the expression:
tan⁡2θ=1021−1016\tan 2\theta = \frac{\frac{\sqrt{10}}{2}}{1 - \frac{10}{16}}
tan⁡2θ=102616\tan 2\theta = \frac{\frac{\sqrt{10}}{2}}{\frac{6}{16}}
tan⁡2θ=102⋅166\tan 2\theta = \frac{\sqrt{10}}{2} \cdot \frac{16}{6}
tan⁡2θ=8106\tan 2\theta = \frac{8\sqrt{10}}{6}
Simplify further:
tan⁡2θ=4103\tan 2\theta = \frac{4\sqrt{10}}{3}
The exact value of tan⁡2θ\tan 2\theta is:
4103\boxed{\frac{4\sqrt{10}}{3}}

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