Question

Хелп plz: sin82°30'*sin37°30' Cos35°sin25° Cos15°cos5°

Answer 1

sin82°30'*sin37°30' = 0.5(cos45-cos120)= 0.5(√2/2 + 0.5)= $\frac{ \sqrt{2} + 1 }{4}$
Cos35°sin25° = 0.5(sin60 + sin 10)=0.5(√3/2 - sin 10 )
Cos15°cos5° = 0.5(cos20 + cos10)

Answer 2

To solve this expression, we can use the trigonometric identities of product-to-sum and sum-to-product.

First, let's use the product-to-sum identity to simplify the expression:

sin(82°30')*sin(37°30') = 1/2 * [cos(82°30' - 37°30') - cos(82°30' + 37°30')] = 1/2 * [cos45° - cos120°] = 1/2 * [sqrt(2)/2 + 1/2] = (sqrt(2) + 1) / 4

cos(35°)*sin(25°) = 1/2 * [sin(35° + 25°) - sin(35° - 25°)] = 1/2 * [sin60° - sin10°] = 1/4 * [sqrt(3) - sin10°]

cos(15°)*cos(5°) = 1/2 * [cos(15° + 5°) + cos(15° - 5°)] = 1/2 * [cos20° + cos10°] = 1/4 * [sqrt(5) + sqrt(2)]

Now we can substitute these values into the original expression and simplify:

(sin82°30'sin37°30') + (cos35°sin25°) + (cos15°*cos5°) = [(sqrt(2) + 1) / 4] + [1/4 * (sqrt(3) - sin10°)] + [1/4 * (sqrt(5) + sqrt(2))]

= (sqrt(2) + sqrt(5) + sqrt(3) + 2 - sin10°) / 4

Therefore, the simplified expression is (sqrt(2) + sqrt(5) + sqrt(3) + 2 - sin10°) / 4.