Question 1152350
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*[Tex \Large \sin \theta = \frac{2}{7} \ \ ] is given


*[Tex \Large \cos \theta = \frac{3\sqrt{5}}{7} \ \ ] is also given


*[Tex \Large \tan \theta = \frac{\sin \theta}{\cos \theta} \ \ ] is one of the many trig identities


*[Tex \Large \tan \theta = \sin \theta \div \cos \theta \ \ ]


*[Tex \Large \tan \theta = \frac{2}{7} \div \frac{3\sqrt{5}}{7} \ \ ] Plug in given values


*[Tex \Large \tan \theta = \frac{2}{7} \times \frac{7}{3\sqrt{5}} \ \ ] Flip the second fraction and multiply


*[Tex \Large \tan \theta = \frac{2*7}{7*3\sqrt{5}} \ \ ]


*[Tex \Large \tan \theta = \frac{2}{3\sqrt{5}} \ \ ]The 7s cancel. See note below.


*[Tex \Large \tan \theta = \frac{2\sqrt{5}}{3\sqrt{5}*\sqrt{5}} \ \ ] Multiply top and bottom by {{{sqrt(5)}}} so we can rationalize the denominator.


*[Tex \Large \tan \theta = \frac{2\sqrt{5}}{3\sqrt{5*5}} \ \ ] Use the rule {{{sqrt(x)*sqrt(y) = sqrt(x*y)}}}


*[Tex \Large \tan \theta = \frac{2\sqrt{5}}{3\sqrt{25}} \ \ ]


*[Tex \Large \tan \theta = \frac{2\sqrt{5}}{3*5} \ \ ]


*[Tex \Large \tan \theta = \frac{2\sqrt{5}}{15} \ \ ] is the final answer


Note: if you are not required to rationalize the denominator, then you can stop at this step.
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