Question

Sin a - sin 3a/cos a - 3a=-ctg 2a

Answer 1

Ответ:

sina-sin3a/cosa-cos3a=-ctg2a

2sin(-a)cos2a/(-2sin(-a)sin2a)=-2sinacos2a/(2sinasin2a)=-cos2a/sin2a=

=-ctg2a

-ctg2a=-ctg2a

Пошаговое объяснение:

Answer 2

To simplify the expression sin(a) - sin(3a) / (cos(a) - 3a) = -cot(2a), let's use trigonometric identities to manipulate it.

First, recall the trigonometric identity for the sine of the difference of two angles:

sin(A - B) = sin(A)cos(B) - cos(A)sin(B)

Now, apply this identity to the numerator:

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

Now, the expression becomes:

(sin(a)cos(3a) - cos(a)sin(3a)) / (cos(a) - 3a) = -cot(2a)

Next, use the product-to-sum identities to simplify the numerator:

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

Now the expression becomes:

sin(4a) / (cos(a) - 3a) = -cot(2a)

Now, let's work with the denominator. Remember that cot(2a) can be expressed as 1/tan(2a), and tan(2a) can be written as sin(2a) / cos(2a). So, the right side of the equation becomes:

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

Now, rewrite the equation as:

sin(4a) / (cos(a) - 3a) = -cos(2a) / sin(2a)

To get rid of the denominators, multiply both sides of the equation by sin(2a)(cos(a) - 3a):

sin(4a) * sin(2a)(cos(a) - 3a) = -cos(2a) * (cos(a) - 3a)

Now, expand and simplify:

sin(4a) * sin(2a) * cos(a) - sin(4a) * sin(2a) * 3a = -cos(2a) * cos(a) + cos(2a) * 3a

Now, apply trigonometric identities:

sin(4a) = 2sin(2a)cos(2a) sin(2a) = 2sin(a)cos(a) cos(2a) = 2cos^2(a) - 1

Substitute these identities into the equation:

2sin(2a)cos(2a) * 2sin(a)cos(a) * cos(a) - 2sin(2a)cos(2a) * 3a = - (2cos^2(a) - 1) * cos(a) + 2cos^2(a) * 3a

Now, simplify:

4sin(a)cos^2(a)cos(2a) - 6a sin(2a)cos(2a) = -2cos(a) + cos(a) + 6a cos^2(a)

Now, apply trigonometric identities again:

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

Substitute these identities:

4sin(a)cos^2(a)(2cos^2(a) - 1) - 6a(2sin(a)cos(a))(2cos^2(a) - 1) = -2cos(a) + cos(a) + 6a cos^2(a)

Now, simplify further:

8sin(a)cos^2(a)cos^2(a) - 4sin(a)cos^2(a) - 12a sin(a)cos^2(a)cos^2(a) + 6a sin(a)cos(a) = -2cos(a) + cos(a) + 6a cos^2(a)

Now, let's simplify and group like terms:

(8cos^4(a) - 4cos^2(a))sin(a) - (12a cos^4(a) - 6a cos^2(a))sin(a) = -cos(a) + cos(a) + 6a cos^2(a)

Factor out sin(a) on the left side:

4cos^2(a)(2cos^2(a) - 1 - 3a)sin(a) = 6a cos^2(a)

Now, divide both sides by 6a cos^2(a) to isolate sin(a):

4cos^2(a)(2cos^2(a) - 1 - 3a)sin(a) / (6a cos^2(a)) = 1

Now, simplify:

(2cos^2(a) - 1 - 3a)sin(a) / (3a cos^2(a)) = 1

This is the simplified form of the given equation.