Oblicz wartość wyrażania...- Zadanie 3.43: Matematyka i przykłady jej zastosowań 3. Zakres podstawowy i rozszerzony

Rozwiązanie

$β \to$	$2 k π + α, k \in Z$ $k \cdot 360^{\circ} + α, k \in Z$
$sin β$	$sin α$
$cos β$	$cos α$
$tg β$	$tg α$

$α \to$ $f.tryg. ↓$	$0^{\circ} = 0$	$90^{\circ} = \frac{π}{2}$	$180^{\circ} = π$	$270^{\circ} = \frac{3 π}{2}$	$360^{\circ} = 2 π$
$sin α$	$0$	$1$	$0$	$- 1$	$0$
$cos α$	$1$	$0$	$- 1$	$0$	$1$
$tg α$	$0$	nie istnieje	$0$	nie istnieje	$0$

$a) sin 405^{\circ} + cos 1125^{\circ} = sin (360^{\circ} + 45^{\circ}) + cos (3 \cdot 360^{\circ} + 45^{\circ}) = sin 45^{\circ} + cos 45^{\circ} = \frac{2}{2} + \frac{2}{2} = 2$

$b) tg 1500^{\circ} \cdot sin 390^{\circ} = tg (4 \cdot 360^{\circ} + 60^{\circ}) \cdot sin (360^{\circ} + 30^{\circ}) = tg 60^{\circ} \cdot sin 30^{\circ} = 3 \cdot \frac{1}{2} = \frac{3}{2}$

$c) cos 450^{\circ} - sin 540^{\circ} = cos (360^{\circ} + 90^{\circ}) - sin (360^{\circ} + 180^{\circ}) = cos 90^{\circ} - sin 180^{\circ} = 0 - 0 = 0$

$d) 3 tg 390^{\circ} + 2 sin 780^{\circ} = 3 tg (360^{\circ} + 30^{\circ}) + 2 sin (2 \cdot 360^{\circ} + 60^{\circ}) = 3 tg 30^{\circ} + 2 sin 60^{\circ} = 3 \cdot \frac{3}{3} + 2 \cdot \frac{3}{2} = 3 + 3 = 23$

$e) cos \frac{33}{4} π + 5 sin \frac{9}{4} π = cos (8 π + \frac{π}{4}) + 5 sin (2 π + \frac{π}{4}) = cos \frac{π}{4} + 5 sin \frac{π}{4} = \frac{2}{2} + 5 \cdot \frac{2}{2} = \frac{6 2}{2} = 32$

$f) tg 21 π - cos \frac{9}{2} π = tg (10 π + π) - cos (4 π + \frac{π}{2}) = tg π - cos \frac{π}{2} = 0 - 0 = 0$

$g) sin \frac{17}{4} π \cdot cos \frac{17}{4} π = sin (4 π + \frac{π}{4}) \cdot cos (4 π + \frac{π}{4}) = sin \frac{π}{4} \cdot cos \frac{π}{4} = \frac{2}{2} \cdot \frac{2}{2} = \frac{1}{2}$

$h) 3 tg \frac{19}{6} π + 4 sin \frac{25}{6} π = 3 tg (2 π + \frac{7}{6} π) + 4 sin (4 π + \frac{π}{6}) = 3 tg (π + \frac{π}{6}) + 4 sin \frac{π}{6} = 3 tg \frac{π}{6} + 4 sin \frac{π}{6} = 3 \cdot \frac{3}{3} + 4 \cdot \frac{1}{2} = 3 + 2$