Korzystając ze wzorów na ...- Zadanie 1: Matematyka z plusem 3. Zakres rozszerzony

Rozwiązanie

Obliczamy wartości danych wyrażeń, korzystając również ze wzorów redukcyjnych ze strony 179.

$a) cos 75^{\circ} = cos (45^{\circ} + 30^{\circ}) = cos 45^{\circ} \cdot cos 30^{\circ} - sin 45^{\circ} \cdot sin 30^{\circ} = \frac{2}{2} \cdot \frac{3}{2} - \frac{2}{2} \cdot \frac{1}{2} = \frac{6}{4} - \frac{2}{4} = \frac{6 - 2}{4}$

$b) sin 105^{\circ} = sin (90^{\circ} + 15^{\circ}) = cos 15^{\circ} = cos (45^{\circ} - 30^{\circ}) = cos 45^{\circ} \cdot cos 30^{\circ} + sin 45^{\circ} \cdot sin 30^{\circ} = \frac{2}{2} \cdot \frac{3}{2} + \frac{2}{2} \cdot \frac{1}{2} = \frac{6}{4} + \frac{2}{4} = \frac{6 + 2}{4}$

$c) sin 15^{\circ} = sin (45^{\circ} - 30^{\circ}) = sin 45^{\circ} \cdot cos 30^{\circ} - cos 45^{\circ} \cdot sin 30^{\circ} = \frac{2}{2} \cdot \frac{3}{2} - \frac{2}{2} \cdot \frac{1}{2} = \frac{6}{4} - \frac{2}{4} = \frac{6 - 2}{4}$

$d) cos (- 15^{\circ}) = cos 15^{\circ} = cos (45^{\circ} - 30^{\circ}) = cos 45^{\circ} \cdot cos 30^{\circ} + sin 45^{\circ} \cdot sin 30^{\circ} = \frac{2}{2} \cdot \frac{3}{2} + \frac{2}{2} \cdot \frac{1}{2} = \frac{6}{4} + \frac{2}{4} = \frac{6 + 2}{4}$

$e) cos (- 105^{\circ}) = cos 105^{\circ} = cos (90^{\circ} + 15^{\circ}) = - sin 15^{\circ} = - sin (45^{\circ} - 30^{\circ}) = - (sin 45^{\circ} \cdot cos 30^{\circ} - cos 45^{\circ} \cdot sin 30^{\circ}) = - (\frac{2}{2} \cdot \frac{3}{2} - \frac{2}{2} \cdot \frac{1}{2}) = - (\frac{6}{4} - \frac{2}{4}) = - \frac{6 - 2}{4} = \frac{- ( 6 - 2 )}{4} = \frac{2 - 6}{4}$

$f) sin 165^{\circ} = sin (180^{\circ} - 15^{\circ}) = sin 15^{\circ} = sin (45^{\circ} - 30^{\circ}) = sin 45^{\circ} \cdot cos 30^{\circ} - cos 45^{\circ} \cdot sin 30^{\circ} = \frac{2}{2} \cdot \frac{3}{2} - \frac{2}{2} \cdot \frac{1}{2} = \frac{6}{4} - \frac{2}{4} = \frac{6 - 2}{4}$

$g) tg 15^{\circ} = tg (45^{\circ} - 30^{\circ}) = \frac{tg 45 ^{\circ} - tg 30 ^{\circ}}{1 + tg 45 ^{\circ} \cdot tg 30 ^{\circ}} = \frac{1 - \frac{3}{3}}{1 + 1 \cdot \frac{3}{3}} = \frac{\frac{3}{3} - \frac{3}{3}}{\frac{3}{3} + \frac{3}{3}} = \frac{\frac{3 - 3}{3}}{\frac{3 + 3}{3}} = \frac{3 - 3}{3 + 3} = \frac{3 - 3}{3 + 3} \cdot \frac{3 - 3}{3 - 3} = \frac{( 3 - 3 ) ( 3 - 3 )}{( 3 + 3 ) ( 3 - 3 )} = \frac{3 ^{2} - 2 \cdot 3 \cdot 3 + ( 3 ) ^{2}}{3 ^{2} - ( 3 ) ^{2}} = \frac{9 - 6 3 + 3}{9 - 3} = \frac{12 - 6 3}{6} = 2 - 3$

$h) tg (- 105^{\circ}) = \frac{sin ( - 105 ^{\circ} )}{cos ( - 105 ^{\circ} )} = \frac{- sin 105 ^{\circ}}{cos 105 ^{\circ}} = \frac{- sin ( 90 ^{\circ} + 15 ^{\circ} )}{cos ( 90 ^{\circ} + 15 ^{\circ} )} = \frac{- cos 15 ^{\circ}}{- sin 15 ^{\circ}} = \frac{cos 15 ^{\circ}}{sin 15 ^{\circ}} = \frac{cos ( 45 ^{\circ} - 30 ^{\circ} )}{sin ( 45 ^{\circ} - 30 ^{\circ} )} = \frac{cos 45 ^{\circ} \cdot cos 30 ^{\circ} + sin 45 ^{\circ} \cdot sin 30 ^{\circ}}{sin 45 ^{\circ} \cdot cos 30 ^{\circ} - cos 45 ^{\circ} \cdot sin 30 ^{\circ}} = \frac{\frac{2}{2} \cdot \frac{3}{2} + \frac{2}{2} \cdot \frac{1}{2}}{\frac{2}{2} \cdot \frac{3}{2} - \frac{2}{2} \cdot \frac{1}{2}} = \frac{\frac{6}{4} + \frac{2}{4}}{\frac{6}{4} - \frac{2}{4}} = \frac{\frac{6 + 2}{4}}{\frac{6 - 2}{4}} = \frac{6 + 2}{6 - 2} = \frac{6 + 2}{6 - 2} \cdot \frac{6 + 2}{6 + 2} = \frac{( 6 + 2 ) ( 6 + 2 )}{( 6 - 2 ) ( 6 + 2 )} = \frac{( 6 ) ^{2} + 2 \cdot 6 \cdot 2 + ( 2 ) ^{2}}{( 6 ) ^{2} - ( 2 ) ^{2}} = \frac{6 + 2 12 + 2}{6 - 2} = \frac{8 + 2 \cdot 2 3}{4} = \frac{8 + 4 3}{4} = 2 + 3$