Trigonometric Equations I - Challenging

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Question 1: Solve the equation

\cos (\frac{π}{8} - 4 x) = 1

x = \frac{π}{32} - \frac{π n}{2}, n \in Ζ

x = \frac{π}{8} - \frac{π n}{2}, n \in Ζ

x = \frac{π}{8} - 2 π n, n \in Ζ

x = \frac{π}{32} - 2 π n, n \in Ζ

Question 2: Find the biggest negative root of the equation

\sin (3 x - \frac{π}{15}) = - 1

x = - π

x = - \frac{13 π}{45}

x = - \frac{13 π}{90}

x = - \frac{17 π}{90}

Question 3: Find the sum of the roots of the equation

t g 2 x = - \sqrt{3}

, which belong to the range

[- π; \frac{π}{2}]

- \frac{π}{3}

- \frac{π}{2}

\frac{π}{6}

- \frac{5 π}{6}

Question 4: Solve the equation

a r c c o s x = \frac{1}{3}

s i n \frac{1}{3}

\frac{π}{3}

c o s \frac{1}{3}

\frac{1}{3}

Question 5: Solve the inequality

a r c t g (4 x - 5) > - \frac{π}{3}

x \in (- 5; \frac{π}{3})

x \in (- \frac{π}{3}; 5)

x \in (\frac{5 - \sqrt{3}}{4}; + \infty)

x \in (- \infty; \frac{5 - \sqrt{3}}{4})

Question 6: Solve the equation

2 s i n x c o s x + \cos^{2} x = 1

x = π n, n \in Ζ

x = \frac{π}{2} + π n, n \in Ζ

[\begin{matrix} x = π n \\ x = a r c t g 2 + π n \end{matrix} n \in Ζ

x = a r c t g 2

Question 7: Solve the equation

c o s x + t g x = 0

x = {(- 1)}^{n} a r c s i n \frac{1 \pm \sqrt{5}}{2} + π n, n \in Ζ

x = a r c s i n \frac{1 - \sqrt{5}}{2} + 2 π n, n \in Ζ

x = \frac{1 \pm \sqrt{5}}{2}

x = {(- 1)}^{n} a r c s i n \frac{1 - \sqrt{5}}{2} + π n, n \in Ζ

Question 8: Solve the equation

\sqrt{c o s 2 x} c o s x = 0

\frac{1}{2}; 1

x = \frac{π}{4} + \frac{π n}{2}, n ϵ Ζ

x = \frac{π}{4} + \frac{π n}{2}; \frac{π}{2} + π n, n ϵ Ζ;

x = \frac{π}{2} + π n, n ϵ Ζ

Question 9: Solve the inequality

s i n x < \frac{\sqrt{3}}{2}

x \in (\frac{2 π}{3} + 2 π n; \frac{7 π}{3} + 2 π n), n \in Ζ

x \in (\frac{π}{3} + 2 π n; \frac{7 π}{3} + 2 π n), n \in Ζ

x \in (\frac{π}{3} + 2 π n; \frac{2 π}{3} + 2 π n), n \in Ζ

x < \frac{π}{3}

Question 10: Solve the equation

s i n 3 x + s i n x = s i n 2 x

x = \frac{π n}{2}, \pm \frac{π}{3} + 2 π n, n ϵ Ζ

x = 2 π n, n ϵ Ζ

x = \pm \frac{π}{3} + 2 π n, n ϵ Ζ

x = \frac{π n}{2}, n ϵ Ζ

Test: Trigonometric Equations I - Challenging