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Question

Mathematics Question on Trigonometric Functions

If tan⁡ θ=cos⁡25∘+sin⁡25∘cos⁡25∘−sin⁡25∘\tan \, \theta = \frac{\cos 25^{\circ} + \sin 25^{\circ} }{\cos 25^{\circ} - \sin 25^{\circ} } and θ\theta is in the third quadrant, then θ=\theta =

A

200∘200^{\circ}

B

205∘205^{\circ}

C

225∘225^{\circ}

D

250∘250^{\circ}

Answer

250∘250^{\circ}

Explanation

Solution

If tan⁡θ=cos⁡25∘+sin⁡25∘cos⁡25∘−sin⁡25∘=1+tan⁡25∘1−tan⁡25∘\tan \theta=\frac{\cos 25^{\circ}+\sin 25^{\circ}}{\cos 25^{\circ}-\sin 25^{\circ}}=\frac{1+\tan 25^{\circ}}{1-\tan 25^{\circ}} =tan⁡(45∘+25∘)=\tan \left(45^{\circ}+25^{\circ}\right) ∵θ\because \theta is in the third quadrant, so tan⁡θ=tan⁡70∘=tan⁡(180∘+70∘)=tan⁡250∘\tan \theta=\tan 70^{\circ}=\tan \left(180^{\circ}+70^{\circ}\right)=\tan 250^{\circ} ⇒θ=250∘.\Rightarrow \theta=250^{\circ} .