31. Вычислите: а) \( 2 \arcsin \left(-\frac{\sqrt{3}}{2}\right)+\operatorname{arctg}(-1)+\arccos \frac{\sqrt{2}}{2} \) б) \( 3 \arcsin \frac{1}{2}+4 \arccos \left(-\frac{1}{\sqrt{2}}\right)-\operatorname{arcctg}(-\sqrt{3}) \) в) \( \operatorname{arctg}(-\sqrt{3})+\arccos \left(-\frac{\sqrt{3}}{2}\right)+\arcsin 1 \) г) \( \arcsin (-1)-\frac{3}{2} \arccos \frac{1}{2}+3 \operatorname{arctg}\left(-\frac{1}{\sqrt{3}}\right) \)

Solve the equation for exact solutions over the interval \( [0,2 \pi) \) \( 2 \tan 3 x=2 \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is \( \left\{\frac{\pi}{12}, \frac{5 \pi}{12}, \frac{3 \pi}{4}, \frac{13 \pi}{12}, \frac{17 \pi}{12}, \frac{7 \pi}{4}\right\} \). (Type an exact answer, using \( \pi \) as needed. Type your answer in radians. Use integers or fractions for any numbers in the expression. Use a com B. The solution is the empty set.

Solve \( \sin ^{2}(t)=-8 \cos (t) \) for all solutions \( 0 \leq t<2 \pi \)

(d) Quando fazemos determinaçōes experimentais de uma grandeza \( X \), representamos seu valor numérico final acompanhado do erro \( \Delta X \) associado à medida, na forma \( X \pm \Delta X \). Para calcularmos o valor de uma funçāo \( f(X \pm \Delta X) \), utilizamos a regra \[ f(X \pm \Delta X)=f(X) \pm f^{\prime}(X) \Delta X \] Obtenha as fórmulas referentes a (i) \( \sin (X \pm \Delta X) \) e (ii) \( \cos (X \pm \Delta X) \)

A pylon \( \$ \$ 23.7 \$ \$ \mathrm{~m} \) high casts a shadow \( \$ \$ 46.8 \$ \$ \mathrm{~m} \) long. Determine the angle of elevation of the sun to the nearest tenth of a degree. 23.7 m high casts a shadow 46.8 m long. Determine the angle of elevation of the sun to the nearest tenth of a degree. Show answer

4) \( \operatorname{Si} \tan \Phi=\frac{\sqrt{6}}{3} \), donde " \( \Phi \) " es un ángulo agudo, calcular: \( \mathrm{P}=\sqrt{3} \operatorname{Sec} \Phi+\sqrt{2} \operatorname{Csc} \Phi \)