Question

Sina-3sin2a+sin3a/cosa-3cos2a+cos3a

Answer 1

Sina-(3*2sina*cosa)/(cosa)-3cos2a+cos3a
sina-(3sin(2a))/(cosa)-3cos2a+cos3a
sina-3*2sina-3cos2a+cos3a
(1-3*2)*sina-3cos2a+cos3a
(1-6)*sina-3cos2a+cos3a

Answer 2

To simplify the expression (Sina-3sin2a+sin3a)/(cosa-3cos2a+cos3a), we'll use trigonometric identities to rewrite the numerator and denominator in a more convenient form.

Step 1: Simplifying the numerator (Sina - 3sin2a + sin3a)

We'll use the sum-to-product identities for sin(A) - sin(B) and sin(A) + sin(B):

1. sin(A) - sin(B) = 2sin((A - B)/2) * cos((A + B)/2)
2. sin(A) + sin(B) = 2sin((A + B)/2) * cos((A - B)/2)

In our expression, A = 3a and B = 2a.

Sina - 3sin2a + sin3a = sin(3a) - 3sin(2a) + sin(3a) [Using the identity sin(A) = sin(-A)] = 2sin(3a) - 3sin(2a)

Step 2: Simplifying the denominator (cosa - 3cos2a + cos3a)

We'll use the sum-to-product identities for cos(A) - cos(B) and cos(A) + cos(B):

1. cos(A) - cos(B) = -2sin((A + B)/2) * sin((A - B)/2)
2. cos(A) + cos(B) = 2cos((A + B)/2) * cos((A - B)/2)

In our expression, A = 3a and B = 2a.

cosa - 3cos2a + cos3a = cos(3a) - 3cos(2a) + cos(3a) [Using the identity cos(A) = cos(-A)] = 2cos(3a) - 3cos(2a)

Now, the original expression becomes:

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

Step 3: Using trigonometric identities to further simplify the expression

We'll use the double-angle identities for sin(2A) and cos(2A):

1. sin(2A) = 2sin(A) * cos(A)
2. cos(2A) = cos^2(A) - sin^2(A)

For our expression, A = 3a.

sin(2A) = sin(6a) = 2sin(3a) * cos(3a)

cos(2A) = cos(6a) = cos^2(3a) - sin^2(3a)

The expression now becomes:

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

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

Step 4: Using the Pythagorean identity sin^2(A) + cos^2(A) = 1

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

Now, the expression becomes:

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

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

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

This is the simplified form of the expression.