Ҳар як масъалаи математикиро ҳал кунед

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1. a = 4,\ b = 6,\ C = 90^\circ

   Two sides and the included angle (SAS): find the third side with the law of cosines.

2. c^2 = a^2 + b^2 - 2ab\cos C = 52

   Law of cosines.

3. c = 2 \sqrt{13} \approx 7.2111

4. P = a + b + c = 2 \sqrt{13} + 10

   Perimeter.

5. s = \tfrac{P}{2} = \sqrt{13} + 5,\quad A = \sqrt{s(s-a)(s-b)(s-c)} = 12

   Heron's formula for the area.

6. \cos A = \frac{b^2 + c^2 - a^2}{2bc} = \frac{3 \sqrt{13}}{13} \Rightarrow A \approx 33.690^\circ

   Law of cosines for angle A (opposite side a).

7. \cos B = \frac{a^2 + c^2 - b^2}{2ac} = \frac{2 \sqrt{13}}{13} \Rightarrow B \approx 56.310^\circ

   Law of cosines for angle B (opposite side b).

8. \cos C = \frac{a^2 + b^2 - c^2}{2ab} = 0 \Rightarrow C \approx 90.000^\circ

   Law of cosines for angle C (opposite side c).

9. A + B + C = 180.0^\circ

   The angles add to 180° — a right, scalene triangle.