Question

Sin225° * cos105° + cos315° * sin285°= Вычислить

Answer 1

Sin(180+45)*cos(90+45)+cos(360-45)*sin(360-75)=sin45*sin45+cos45*-sin30=1+cos45*-sin30=1+ $\sqrt{2} /2$*(-1/2)=-0.8 $\sqrt{2} /2$

sin285= sin(360-75)=-sin75= -sin(45+30) 

Answer 2

To compute the value of the expression, we'll use the trigonometric identities:

1. sin(225°) = -sin(45°) = -√2 / 2
2. cos(105°) = cos(180° - 75°) = -cos(75°) = -(√6 + √2) / 4
3. cos(315°) = cos(360° - 45°) = cos(45°) = √2 / 2
4. sin(285°) = sin(360° - 75°) = -sin(75°) = - (√6 - √2) / 4

Now, let's substitute these values into the expression:

sin(225°) * cos(105°) + cos(315°) * sin(285°) = (-√2 / 2) * (-(√6 + √2) / 4) + (√2 / 2) * (-(√6 - √2) / 4)

Now, simplify the expression step by step:

1. (-√2 / 2) * (-(√6 + √2) / 4) = (√2 / 2) * (√6 + √2) / 4 = (√2 * √6 + 2) / 8 = (√12 + 2) / 8
2. (√2 / 2) * (-(√6 - √2) / 4) = -(√2 * √6 - √2) / 8 = (-√12 + √2) / 8

Now, the expression becomes:

(√12 + 2) / 8 + (-√12 + √2) / 8

Combine the fractions:

(√12 - √12 + 2) / 8 = (2) / 8 = 1 / 4

So, the value of the expression is 1/4.