Question

3. Se cos x = -1/4 e π/2 ≤ a ≤ π, determine sen 2x e cos 2x.

Answer 1

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$\boxed{\begin{array}{l}\sf cos(x)=-\dfrac{1}{4}\implies cos^2(x)=\dfrac{1}{16}\\\sf sen^2(x)=\dfrac{16}{16}-\dfrac{1}{16}=\dfrac{15}{16}\\\sf sen(x)=\sqrt{\dfrac{15}{16}}=\dfrac{\sqrt{15}}{4}\\\sf sen(2x)=2\cdot sen(x)\cdot cos(x)\\\sf sen(2x)=2\cdot\bigg(\dfrac{\sqrt{15}}{4}\bigg)\cdot\bigg(-\dfrac{1}{4}\bigg)\\\sf sen(2x)=-\dfrac{\sqrt{15}}{8}\\\sf cos(2x)=2cos^2(x)-1\\\sf cos(2x)=2\cdot\bigg(\dfrac{1}{16}\bigg)-1\\\sf cos(2x)=\dfrac{1}{8}-1=-\dfrac{7}{8}\end{array}}$