If cos⁡(θ)=−23 and 450∘&lt;θ&lt;540∘, find1. The exact value of cos⁡(12θ)2.The exact value of tan⁡(2θ)

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If cos⁡(θ)=−23 and 450∘&amp;lt;θ&amp;lt;540∘, find1. The exact value of cos⁡(12θ)2.The exact value of tan⁡(2θ)

Aiyana Richmond · Accepted Answer

You were given that cos⁡θ=−23, with 450∘&amp;lt;θ&amp;lt;540∘Recall thatcos⁡(θ2)=±1+cos⁡θ2where the sign is determined by the measure of angle θ2. Since 450∘&amp;lt;θ&amp;lt;540∘, we may conclude that 225∘&amp;lt;θ2&amp;lt;270∘, so θ2 is a third-quadrant angle. Hence, its cosine is negative. Thus,cos⁡(θ2)=−1+cos⁡θ2=−1−232=−132=−16so you found the correct magnitude but did not take into account into the sign of cos⁡(θ2).To determine tan⁡(2θ), we can use the formulatan⁡(2θ)=sin⁡(2θ)cos⁡(2θ)together with the double angle formulassin⁡(2θ)=2sin⁡θcos⁡θcos⁡(2θ)=cos2θ−sin2θ=2cos2θ−1=1−2sin2θSince 450∘&amp;lt;θ&amp;lt;540∘, θ is a second-quadrant angle, so sin⁡θ&amp;gt;0. Hence,sin⁡θ=1−cos2θ=1−(−23)2=1−49=59=53Therefore,tan⁡(2θ)=sin⁡(2θ)cos⁡(2θ)=2sin⁡θcos⁡θcos2θ−sin2θ=2(53)(−23)(−23)2−(53)2=−45949−59=−454−5=45

If \cos(\theta)=-\frac{2}{3} and 450^\circ<\theta<540^\circ, find 1. The exact value of \cos(\frac{1}{2}\theta) 2.The

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