Question

Вычеслить sin75 X cos 15 - sin 15 X cos 75

Answer 1

=sin(75-15)=sin 60=√3/2

Answer 2

To calculate sin(75) × cos(15) - sin(15) × cos(75), we'll first express the angles in terms of simpler angles using trigonometric identities.

1. sin(75) can be expressed as sin(45 + 30) using the sum-to-product identity: sin(a + b) = sin(a) × cos(b) + cos(a) × sin(b). Therefore, sin(75) = sin(45 + 30) = sin(45) × cos(30) + cos(45) × sin(30).

2. cos(15) can be expressed as cos(45 - 30) using the difference-to-product identity: cos(a - b) = cos(a) × cos(b) + sin(a) × sin(b). Therefore, cos(15) = cos(45 - 30) = cos(45) × cos(30) + sin(45) × sin(30).

3. sin(15) can be expressed as sin(45 - 30) using the difference-to-product identity: sin(a - b) = sin(a) × cos(b) - cos(a) × sin(b). Therefore, sin(15) = sin(45 - 30) = sin(45) × cos(30) - cos(45) × sin(30).

4. cos(75) can be expressed as cos(45 + 30) using the sum-to-product identity: cos(a + b) = cos(a) × cos(b) - sin(a) × sin(b). Therefore, cos(75) = cos(45 + 30) = cos(45) × cos(30) - sin(45) × sin(30).

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

sin(75) × cos(15) - sin(15) × cos(75) = (sin(45) × cos(30) + cos(45) × sin(30)) × (cos(45) × cos(30) + sin(45) × sin(30)) - (sin(45) × cos(30) - cos(45) × sin(30)) × (cos(45) × cos(30) - sin(45) × sin(30))

Now, you can simplify the expression and compute the result. Keep in mind that sin(45) and cos(45) are constants that you can look up in trigonometric tables or use a calculator.