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Question: If \(\tan\alpha = \frac{1}{7}\) and \(\sin\beta = \frac{1}{\sqrt{10}}\), then \(\tan(\alpha + 2\beta...

If tan⁡α=17\tan\alpha = \frac{1}{7} and sin⁡β=110\sin\beta = \frac{1}{\sqrt{10}}, then tan⁡(α+2β\tan(\alpha + 2\beta) is qual to

A

1

B

0

C

12\frac{1}{2}

D

34\frac{3}{4}

Answer

1

Explanation

Solution

tan⁡α=17,sin⁡β=110\tan\alpha = \frac{1}{7},\sin\beta = \frac{1}{\sqrt{10}} ⇒ tan⁡β=13\tan\beta = \frac{1}{3} ⇒ tan⁡2β\tan 2\beta = 231−19=34\frac{\frac{2}{3}}{1 - \frac{1}{9}} = \frac{3}{4}

∴tan⁡(α+2β)=17+341−328=4+2125=1\tan(\alpha + 2\beta) = \frac{\frac{1}{7} + \frac{3}{4}}{1 - \frac{3}{28}} = \frac{4 + 21}{25} = 1