# Trigonometry Problems

Exercise 1: Prove that:
A) $\beta - \alpha = \frac{\pi}{4}$, if $\sin \alpha = \frac{\sqrt{5}}{5}$, $\alpha \in$ I quadrant, $\cos \beta = \frac{\sqrt{10}}{10}$, $\beta \in$ I quadrant.

B) $\alpha + \beta = \frac{\pi}{2}$, if $\sin \alpha = \frac{15}{17}$, $\alpha \in$ I quadrant, $\cos \beta = \frac{8}{17}$, $\beta \in$ I quadrant.

C) $\alpha + \beta + \gamma = \frac{\pi}{2}$, if $\sin \alpha = \frac{1}{3}$, $\sin \beta = \frac{\sqrt{11}}{33}$ and $\sin \gamma = \frac{3\sqrt{11}}{11}$, where $\alpha, \beta, \gamma \in$ I quadrant.

Exercise 2: Prove that if $\alpha + \beta + \gamma = \frac{\pi}{2}$, then
A) $\tan \alpha \cdot \tan \beta + \tan\beta \cdot \tan\gamma + \tan\alpha\cdot \tan\gamma = 1$

B) $\cot\alpha + \cot\beta + \cot \beta = \cot\alpha \cdot \cot\beta \cdot \cot \gamma$

Exercise 3: Prove that if
$\alpha + \beta = \frac{\pi}{4}$

then $(1 + \tan\alpha)(1 + \tan\beta) = 2$.

Exercise 4: Prove that
A) $\cos\alpha + \sqrt{3}\sin\alpha \le 2$

B) $\sqrt{2}(\sin\alpha + \cos\alpha) \le 2$

C) $\sin2\alpha\cos\beta + \cos2\alpha\sin\beta \le 1$

D) $\sin(\alpha + \beta) \le \sin\alpha + \sin\beta$

E) $\cos(\alpha + \beta) < \cos\alpha - \sin\alpha\cdot\sin\beta$

F) $\sqrt{3}\cos\alpha - \sin\alpha \le 2$

G) $\sin\alpha - \cos\alpha \le \sqrt{2}$

Exercise 5: Find the maximum value of $y = \sqrt{3}\sin2x - \cos2x$ and the value of $x$, if $0 \le x \le \pi$.

Exercise 6: Calculate:
A) $\sin67^\circ \cdot \cos68^\circ - \cos67^\circ\cdot \sin68^\circ$

B) $\sin27^\circ \cdot cos33^\circ + \sin63^\circ\cdot cos57^\circ + \cos30^\circ$

C) $\frac{\tan66^\circ \cdot \cot36^\circ - 1}{\tan66^\circ + \cot36^\circ }$

D) $\frac{\tan92^\circ + \cot2^\circ }{1 - \tan92^\circ \cdot \cot2^\circ}$

E) $\cos1^\circ + \cos121^\circ + \cos241^\circ$

F) $\frac{\sin110^\circ \cdot \sin250^\circ + \cos540^\circ \cdot cos290^\circ }{\cos1260^\circ }$

Exercise 7: Prove the following identities:
A) $\frac{1}{2}(\cos \alpha + \sqrt{3}\sin \alpha ) = \sin(30^\circ + \alpha)$

B) $\cos\alpha - \sqrt{3}\sin\alpha = 2\cos(60^\circ + \alpha)$

C) $\tan20^\circ + \tan23^\circ + \tan20^\circ \cdot \tan25^\circ = 1$

D) $\tan20^\circ + \tan40^\circ + \sqrt{3}\tan20^\circ \cdot \tan40^\circ = \sqrt{3}$

E) $\sin(\alpha + \beta)\cdot \cos(\alpha - \beta) = \sin^2\alpha - \sin^2\beta$

F) $\cos(\alpha + \beta)\cdot \cos(\alpha - \beta) = \cos^2\alpha - \cos^2\beta$

G) $\sin^2\left(\alpha - \frac{\pi}{6} \right) + \sin^2\left(\alpha + \frac{\pi}{6} \right) - \sin^2 \alpha = \frac{1}{2}$

Feedback   Contact email: