Question

Sin 2n/5 * cos 7n/30 - sin 7n/30 * cos 2n/5​

Answer 1

Ответ:

sin 2n/5 * cos 7n/30 - sin 7n/30 * cos 2n/5=sin2n*cos7n/150 - sin7n sin2n/150=sin2ncos7n-sin7n-sin2n/150.

Answer 2

To simplify the expression sin(2n/5) * cos(7n/30) - sin(7n/30) * cos(2n/5), we can use trigonometric identities.

First, let's recall the double angle and half-angle identities:

1. sin(2θ) = 2sin(θ)cos(θ)
2. cos(2θ) = cos^2(θ) - sin^2(θ)

Now, we can rewrite the given expression:

sin(2n/5) * cos(7n/30) - sin(7n/30) * cos(2n/5)

Using the double angle identity for sin(2θ), we have:

2sin(n/5)cos(n/5) * cos(7n/30) - sin(7n/30) * cos(2n/5)

Now, let's use the double angle identity for cos(2θ):

cos(2n/5) = cos^2(n/5) - sin^2(n/5)

So, the expression becomes:

2sin(n/5)cos(n/5) * cos(7n/30) - sin(7n/30) * [cos^2(n/5) - sin^2(n/5)]

Now, we can use the identity sin^2(θ) + cos^2(θ) = 1 to simplify:

2sin(n/5)cos(n/5) * cos(7n/30) - sin(7n/30) * (1 - cos^2(n/5))

Now, distribute sin(7n/30) to both terms in the second part:

2sin(n/5)cos(n/5) * cos(7n/30) - sin(7n/30) + sin(7n/30) * cos^2(n/5)

You can leave the expression in this form, and it represents the simplified version of the given expression.