Question

Sin79 sin49 + cos79 cos49

Answer 1

Cos30. Ответ вот.....

Answer 2

We can use the trigonometric identity:

cos(a - b) = cos(a) cos(b) + sin(a) sin(b)

to simplify the expression. Let's rewrite sin 79 and cos 79 in terms of sin 49 and cos 49:

sin 79 = sin (49 + 30) = sin 49 cos 30 + cos 49 sin 30

cos 79 = cos (49 + 30) = cos 49 cos 30 - sin 49 sin 30

Now we can substitute these expressions into the original expression:

sin 79 sin 49 + cos 79 cos 49 = (sin 49 cos 30 + cos 49 sin 30) sin 49 + (cos 49 cos 30 - sin 49 sin 30) cos 49 = sin^2 49 cos 30 + cos^2 49 cos 30 = cos 30 cos^2 49 + sin^2 49 cos 30

We can use the identity cos^2(x) + sin^2(x) = 1 to simplify further:

cos 30 cos^2 49 + sin^2 49 cos 30 = cos 30 (1 - sin^2 49) + sin^2 49 cos 30 = cos 30 - cos 30 sin^2 49 + sin^2 49 cos 30 = cos 30 + sin^2 49 cos 60

Finally, we can use the fact that cos 30 = sqrt(3)/2 and cos 60 = 1/2 to get:

cos 30 + sin^2 49 cos 60 = sqrt(3)/2 + sin^2 49 (1/2) = sqrt(3)/2 + (1 - cos^2 49)/2 = (sqrt(3) + 1)/2 - (cos^2 49)/2

Therefore, the simplified expression is:

(sin 79 sin 49 + cos 79 cos 49) = (sqrt(3) + 1)/2 - (cos^2 49)/2.