Giải các bài toán về giá trị và biểu thức trong toán học

Trong bài viết này, chúng ta sẽ giải hai bài toán liên quan đến giá trị và biểu thức trong toán học. Hai câu hỏi được đưa ra là: Câu 1: Giá trị của \( \cot \frac{95 \pi}{6} \) là bao nhiêu? Câu 2: Giá trị của biểu thức \( P=\sin \left(a-19^{\circ}\right) \cdot \cos \left(a+11^{\circ}\right)-\cos \left(a-19^{\circ}\right) \cdot \sin \left(a+11^{\circ}\right) \) là gì? Đầu tiên, chúng ta sẽ giải câu hỏi thứ nhất. Để tính giá trị của \( \cot \frac{95 \pi}{6} \), chúng ta cần biết rằng \( \cot \theta \) là nghịch đảo của \( \tan \theta \). Vì vậy, chúng ta cần tính giá trị của \( \tan \frac{95 \pi}{6} \) trước. Để tính giá trị này, chúng ta có thể sử dụng các quy tắc và công thức trong toán học. Đầu tiên, chúng ta biết rằng \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Vì vậy, chúng ta cần tính giá trị của \( \sin \frac{95 \pi}{6} \) và \( \cos \frac{95 \pi}{6} \). Theo công thức, chúng ta có thể tính được \( \sin \frac{95 \pi}{6} = \sin \frac{\pi}{6} = \frac{1}{2} \) và \( \cos \frac{95 \pi}{6} = \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \). Sau đó, chúng ta tính được \( \tan \frac{95 \pi}{6} = \frac{\sin \frac{95 \pi}{6}}{\cos \frac{95 \pi}{6}} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \). Vậy, giá trị của \( \cot \frac{95 \pi}{6} \) là \( \frac{\sqrt{3}}{3} \). Tiếp theo, chúng ta sẽ giải câu hỏi thứ hai. Để tính giá trị của biểu thức \( P \), chúng ta cần sử dụng các quy tắc và công thức trong toán học. Đầu tiên, chúng ta có thể sử dụng công thức \( \sin (a - b) = \sin a \cos b - \cos a \sin b \) và \( \cos (a - b) = \cos a \cos b + \sin a \sin b \). Áp dụng công thức này vào biểu thức \( P \), chúng ta có thể tính được: \( P = \sin \left(a-19^{\circ}\right) \cdot \cos \left(a+11^{\circ}\right)-\cos \left(a-19^{\circ}\right) \cdot \sin \left(a+11^{\circ}\right) \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \) \( = \sin a \cos 19^{\circ} - \cos a \sin 19^{\circ} \cdot \cos a \cos 11^{\circ} + \sin a \sin 11^{\circ} \