Mathematics

Mathematics

Problem

Let P(cos⁡θ, sin⁡θ)P(\cos \theta,\,\sin \theta) be the point on the unit circle corresponding to an angle θ\theta. a. (i) State the coordinates of the point QQ on the unit circle that corresponds to the angle θ+180∘\theta + 180^{\circ} in terms of cos⁡θ\cos \theta and sin⁡θ\sin \theta. b. (ii) Using the coordinates in part (i), show that sin⁡(θ+180∘)=−sin⁡θ\sin(\theta + 180^{\circ}) = -\sin \theta and cos⁡(θ+180∘)=−cos⁡θ\cos(\theta + 180^{\circ}) = -\cos \theta for all real θ\theta. c. (iii) Hence prove that tan⁡(θ+180∘)=tan⁡θ\tan(\theta + 180^{\circ}) = \tan \theta and cot⁡(θ+180∘)=cot⁡θ\cot(\theta + 180^{\circ}) = \cot \theta wherever the functions are defined.

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