1. **Problem:** Given a right triangle \(\triangle ACB\) with side \(b=12\) and angle \(\angle A=12^\circ\), find the missing parts (sides \(a\), \(c\) and angle \(\angle B\)). 2. **Formula and rules:** - In a right triangle, \(\angle C = 90^\circ\). - The sum of angles in a triangle is \(180^\circ\), so \(\angle B = 90^\circ - \angle A\). - Use trigonometric ratios: \(\sin, \cos, \tan\) to find missing sides. - Hypotenuse \(c\) is opposite the right angle. 3. **Step-by-step solution:** - Calculate \(\angle B\): $$\angle B = 90^\circ - 12^\circ = 78^\circ$$ - Use \(\tan\) to find side \(a\): $$\tan(\angle A) = \frac{a}{b} \Rightarrow a = b \times \tan(12^\circ)$$ $$a = 12 \times \tan(12^\circ) \approx 12 \times 0.2126 = 2.55$$ - Use Pythagoras theorem to find hypotenuse \(c\): $$c = \sqrt{a^2 + b^2} = \sqrt{2.55^2 + 12^2} = \sqrt{6.50 + 144} = \sqrt{150.5} \approx 12.27$$ 4. **Answer:** - \(a \approx 2.55\) - \(c \approx 12.27\) - \(\angle B = 78^\circ\) This completes the solution for the first problem.