Esercizio 21 ripasso goniometria e trigonometria

Ripasso goniometria e trigonometria

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Esercizio 21   (\bigstar\largewhitestar\largewhitestar\largewhitestar\largewhitestar). Calcolare il valore delle seguenti espressioni, ricorrendo alle relazioni tra archi associati:

    \[\begin{aligned} &(1)\quad \left(\tan\frac{2\pi}{3}-6\cot\frac{7\pi}{6}\right)\left(\sin\frac{2\pi}{3}+\cos\frac{\pi}{6}\right)-12;\\ &(2)\quad \left(\sin\frac{3\pi}{4}-\sin\left(-\frac{\pi}{4}\right)\right):\left(\tan\frac{5\pi}{4}-\cot\frac{3\pi}{4}\right). \end{aligned}\]

 

Svolgimento. 

(1) Si ha

    \[\begin{aligned} &\left(\tan\frac{2\pi}{3}-6\cot\frac{7\pi}{6}\right)\left(\sin\frac{2\pi}{3}+\cos\frac{\pi}{6}\right)-12=\\ &=\left[\tan\left(\pi-\frac{\pi}{3}\right)-6\cot\left(\pi+\frac{\pi}{6}\right)\right] \left[\sin\left(\pi-\frac{\pi}{3}\right)+\cos\frac{\pi}{6}\right]-12=\\ &=\left(-\tan\frac{\pi}{3}-6\cot\frac{\pi}{6}\right)\left(\sin\frac{\pi}{3}+\cos\frac{\pi}{6}\right)-12=\\ &=\left(-\sqrt{3}-6\sqrt{3}\right)\left(\frac{\sqrt{3}}{2}+\frac{\sqrt{3}}{2}\right)-12 =-7\sqrt{3}\cdot\sqrt{3}-12=-33. \end{aligned}\]

(2) Si ha

    \[\begin{aligned} &\left[\sin\frac{3\pi}{4}-\sin\left(-\frac{\pi}{4}\right)\right]:\left(\tan\frac{5\pi}{4}-\cot\frac{3\pi}{4}\right)=\\ &=\left[\sin\left(\pi-\frac{\pi}{4}\right)-\sin\left(-\frac{\pi}{4}\right)\right]: \left[\tan\left(\pi+\frac{\pi}{4}\right)-\cot\left(\pi-\frac{\pi}{4}\right)\right]=\\ &=\left[\sin\frac{\pi}{4}+\sin\frac{\pi}{4}\right]:\left[\tan\frac{\pi}{4}+\cot\frac{\pi}{4}\right]= \frac{\sqrt{2}}{2}. \end{aligned}\]

Fonte: ignota.