# Test: Trigonometric Equations I - Challenging

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Question 1:   Solve the equation $\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{\pi }{8}-4x\right)=1\text{\hspace{0.17em}}$ .
Question 2:   Find the biggest negative root of the equation $\text{\hspace{0.17em}}\mathrm{sin}\left(3x-\frac{\pi }{15}\right)=-1\text{\hspace{0.17em}}$ .
$x=-\pi$
$x=-\frac{17\pi }{90}$
$x=-\frac{13\pi }{90}$
$x=-\frac{13\pi }{45}$
Question 3:   Find the sum of the roots of the equation $\text{\hspace{0.17em}}tg2x=-\sqrt{3}\text{\hspace{0.17em}}$ , which belong to the range$\text{\hspace{0.17em}}\left[-\pi ;\frac{\pi }{2}\right]\text{\hspace{0.17em}}$ .
$-\frac{5\pi }{6}$
$\frac{\pi }{6}$
$-\frac{\pi }{2}$
$-\frac{\pi }{3}$
Question 4:   Solve the equation $\text{\hspace{0.17em}}arccosx=\frac{1}{3}\text{\hspace{0.17em}}$ .
$\frac{\pi }{3}$
$sin\frac{1}{3}$
Question 5:   Solve the inequality $\text{\hspace{0.17em}}arctg\left(4x-5\right)>-\frac{\pi }{3}\text{\hspace{0.17em}}$ .
$x\in \left(-\frac{\pi }{3};5\right)$
$x\in \left(-5;\frac{\pi }{3}\right)$
$x\in \left(-\infty ;\frac{5-\sqrt{3}}{4}\right)$
$x\in \left(\frac{5-\sqrt{3}}{4};+\infty \right)$
Question 6:   Solve the equation $\text{\hspace{0.17em}}2sinxcosx+{\mathrm{cos}}^{2}x=1\text{\hspace{0.17em}}$ .
$x=arctg2$
Question 7:   Solve the equation $\text{\hspace{0.17em}}cosx+tgx=0\text{\hspace{0.17em}}$ .
$x=\frac{1±\sqrt{5}}{2}$
Question 8:   Solve the equation .
$\frac{1}{2};\text{\hspace{0.17em}}1$
Question 9:   Solve the inequality $\text{\hspace{0.17em}}sinx<\frac{\sqrt{3}}{2}\text{\hspace{0.17em}}$ .
$x<\frac{\pi }{3}$
Question 10:   Solve the equation $\text{\hspace{0.17em}}sin3x+sinx=sin2x\text{\hspace{0.17em}}$ .