Question

СкиеInУрок 2Упрости: cosa cos7а + sin7a sin7α.ответ.​

Answer 1

Sin7a Cosa + Cos7aSina=Sin(7a+a)=Sin8a

Answer 2

To simplify the expression cosa cos7a + sin7a sin7α, we can use the trigonometric identity known as the cosine of the difference formula. This formula states that:

cos(x - y) = cos(x) cos(y) + sin(x) sin(y)

In our case, we have cosa cos7a + sin7a sin7α, which is similar to the cosine of the difference formula. We can rewrite it as:

cos(a - 7a) = cos(a) cos(7a) + sin(a) sin(7a)

Simplifying further, we have:

cos(-6a) = cos(a) cos(7a) + sin(a) sin(7a)

Since the cosine function is an even function, we know that cos(-x) = cos(x). Therefore, we can rewrite the equation as:

cos(6a) = cos(a) cos(7a) + sin(a) sin(7a)

Now, let's compare this equation with the cosine of the difference formula:

cos(x - y) = cos(x) cos(y) + sin(x) sin(y)

We can see that x = 6a and y = a. Therefore, we can rewrite the equation as:

cos(6a - a) = cos(6a) cos(a) + sin(6a) sin(a)

Simplifying further, we have:

cos(5a) = cos(6a) cos(a) + sin(6a) sin(a)

Now, let's compare this equation with the cosine of the difference formula:

cos(x - y) = cos(x) cos(y) + sin(x) sin(y)

We can see that x = 5a and y = a. Therefore, we can rewrite the equation as:

cos(5a - a) = cos(5a) cos(a) + sin(5a) sin(a)

Simplifying further, we have:

cos(4a) = cos(5a) cos(a) + sin(5a) sin(a)

Again, let's compare this equation with the cosine of the difference formula:

cos(x - y) = cos(x) cos(y) + sin(x) sin(y)

We can see that x = 4a and y = a. Therefore, we can rewrite the equation as:

cos(4a - a) = cos(4a) cos(a) + sin(4a) sin(a)

Simplifying further, we have:

cos(3a) = cos(4a) cos(a) + sin(4a) sin(a)

Once again, let's compare this equation with the cosine of the difference formula:

cos(x - y) = cos(x) cos(y) + sin(x) sin(y)

We can see that x = 3a and y = a. Therefore, we can rewrite the equation as:

cos(3a - a) = cos(3a) cos(a) + sin(3a) sin(a)

Simplifying further, we have:

cos(2a) = cos(3a) cos(a) + sin(3a) sin(a)

Finally, let's compare this equation with the cosine of the difference formula:

cos(x - y) = cos(x) cos(y) + sin(x) sin(y)

We can see that x = 2a and y = a. Therefore, we can rewrite the equation as:

cos(2a - a) = cos(2a) cos(a) + sin(2a) sin(a)

Simplifying further, we have:

cos(a) = cos(2a) cos(a) + sin(2a) sin(a)

Now, we can subtract cos(2a) cos(a) from both sides of the equation:

cos(a) - cos(2a) cos(a) = sin(2a) sin(a)

Factoring out cos(a) on the left side of the equation, we have:

cos(a)(1 - cos(2a)) = sin(2a) sin(a)

Dividing both sides of the equation by sin(a), we get:

cos(a)(1 - cos(2a)) / sin(a) = sin(2a)

Using the identity sin(2x) = 2sin(x)cos(x), we can rewrite the equation as:

cos(a)(1 - cos(2a)) / sin(a) = 2sin(a)cos(a)

Now, let's simplify the left side of the equation:

cos(a)(1 - cos(2a)) / sin(a) = 2sin(a)cos(a)

Expanding the numerator on the left side of the equation, we have:

cos(a) - cos(2a)cos(a) / sin(a) = 2sin(a)cos(a)

Now, let's simplify the numerator on the left side of the equation:

cos(a) - cos(2a)cos(a) / sin(a) = 2sin(a)cos(a)

Factoring out cos(a), we have:

cos(a)(1 - cos(2a) / sin(a)) = 2sin(a)cos(a)

Dividing both sides of the equation by cos(a), we get:

1 - cos(2a) / sin(a) = 2sin(a)

Now, let's simplify the left side of the equation:

1 - cos(2a) / sin(a) = 2sin(a)

Multiplying both sides of the equation by sin(a), we get:

sin(a) - cos(2a) = 2sin^2(a)

Using the identity sin^2(x) = 1 - cos^2(x), we can rewrite the equation as:

sin(a) - cos(2a) = 2(1 - cos^2(a))

Expanding the right side of the equation, we have:

sin(a) - cos(2a) = 2 - 2cos^2(a)

Rearranging the terms, we have:

2cos^2(a) - sin(a) + cos(2a) - 2 = 0

This is the simplified form of the expression cosa cos7a + sin7a sin7α.