1317 Вправа 1317Геометрія 8 клас Бурда 2025 · Повторення · стор. 257

35. a і b — катети прямокутного трикутника, c — гіпотенуза, A і B — гострі кути. Заповніть таблицю 42.

c2 = a2 + b2; ∠A + ∠B = 90°;

sin A = cos B = ${\displaystyle \frac{\mathbf{a}}{\mathbf{c}}}$; cos A = sin B = ${\displaystyle \frac{\mathbf{b}}{\mathbf{c}}}$; tg A = ${\displaystyle \frac{\mathbf{a}}{\mathbf{b}}}$.

1) ∠B = 90° – 40° = 50°;

a = 13,5 · sin 40° ≈ 8,68;

b = 13,5 · cos 40° ≈ 10,34.

2) ∠A = 90° – 32° = 58°;

a = 0,25 · cos 32° ≈ 0,21;

b = 0,25 · sin 32° ≈ 0,13.

3) ∠B = 90° – 67° = 23°;

c = ${\displaystyle \frac{\mathbf{47}\mathbf{,}\mathbf{2}}{\mathbf{sin}{\mathbf{67}}^{\mathbf{\circ }}}}$ ≈ 51,28;

b = ${\displaystyle \frac{\mathbf{47}\mathbf{,}\mathbf{2}}{\mathbf{tg}{\mathbf{67}}^{\mathbf{\circ }}}}$ ≈ 20,04.

4) ∠A = 90° – 53° = 37°;

b = 10,2 · tg 53° ≈ 13,54;

c = ${\displaystyle \frac{\mathbf{10}\mathbf{,}\mathbf{2}}{\mathbf{cos}{\mathbf{53}}^{\mathbf{\circ }}}}$ ≈ 16,95.

5) b = ${\displaystyle \sqrt{{\mathbf{20}}^{\mathbf{2}}{\mathbf{-}\mathbf{17}}^{\mathbf{2}}}}$ = ${\displaystyle \sqrt{\mathbf{111}}}$ ≈ 10,54;

sin A = ${\displaystyle \frac{\mathbf{17}}{\mathbf{20}}}$ = 0,85; ∠A ≈ 58°13';

∠B ≈ 90° – 58°13' = 31°47'.

6) ∠B = 90° – 52° = 38°;

a = 75 · tg 52° ≈ 96,0;

c = ${\displaystyle \frac{\mathbf{75}}{\mathbf{cos}{\mathbf{52}}^{\mathbf{\circ }}}}$ ≈ 121,8.

7) c = ${\displaystyle \sqrt{{\mathbf{12}\mathbf{,}\mathbf{7}}^{\mathbf{2}}{\mathbf{+}\mathbf{13}\mathbf{,}\mathbf{4}}^{\mathbf{2}}}}$ = ${\displaystyle \sqrt{\mathbf{340}\mathbf{,}\mathbf{85}}}$ ≈ 18,46;

tg A = ${\displaystyle \frac{\mathbf{12}\mathbf{,}\mathbf{7}}{\mathbf{13}\mathbf{,}\mathbf{4}}}$ ≈ 0,948; ∠A ≈ 43°28';

∠B ≈ 90° – 43°28' = 46°32'.

8) a = ${\displaystyle \sqrt{{\mathbf{12}}^{\mathbf{2}}{\mathbf{-}\mathbf{8}}^{\mathbf{2}}}}$ = ${\displaystyle \sqrt{\mathbf{80}}}$ ≈ 8,94;

sin B = ${\displaystyle \frac{\mathbf{8}}{\mathbf{12}}}$ ≈ 0,667; ∠B ≈ 41°49';

∠A ≈ 90° – 41°49' = 48°11'.