Question

ПОЖАЛЙСТА!!!с   дано:cost=3\\4; 0<t<п\\2 Найти: cos t\\2,  sin t\\2,  tg t\\2 и  ctg t\\2.

Answer 1

cost = 3/4

 

cos(t/2) = ((1 + cost)/2)^(1/2) = ((1 + 3/4)/2)^(1/2) = (7/8)^(1/2) = (7^(1/2))/(2*(2^(1/2)))

 

sin(t/2) = (1 - cos^2(t/2))^(1/2) = (1 - 7/8)^(1/2) = 1/(8^(1/2)) = 1/(2*(2^(1/2)))

 

tg(t/2) = sin(t/2)/cos(t/2) = (1/(2*(2^(1/2))))/((7^(1/2))/(2*(2^(1/2)))) = 1/(7^(1/2))

 

ctg(t/2) = 1/tg(t/2) = 7^(1/2)

 

 

Answer 2

Используем формулы наполовину угла:
$$\cos \frac{t}{2} = \sqrt{\frac{1+\cos t}{2}}=\sqrt{\frac{1+3/4}{2}}=\frac{\sqrt{2}}{2}$$
$$\sin \frac{t}{2} = \sqrt{\frac{1-\cos t}{2}}=\sqrt{\frac{1-3/4}{2}}=\frac{1}{2}$$
$$\tan \frac{t}{2} = \frac{\sin t}{1+\cos t}=\frac{2\sin\frac{t}{2}\cos\frac{t}{2}}{1+2\cos^2\frac{t}{2}-1}=\frac{2\sin\frac{t}{2}\cos\frac{t}{2}}{2\cos^2\frac{t}{2}}=\frac{\sin t}{1+\cos t}=\frac{1}{\sqrt{2}}$$
$$\cot \frac{t}{2}=\frac{1}{\tan\frac{t}{2}}=\sqrt{2}$$