Question

Cos 3 b +cos 5 b + cos b/sin b +sin 5 b +sin 3 b​

Answer 1

Ответ:

ctg3b

Пошаговое объяснение:

sina+sinb=2sin((a+b)/2)cos((a-b)/2)

cosa+cosb=2cos((a+b)/2)cos((a-b)/2)

cosb+cos5b=2cos3bcos2b

sin5b+sinb=2sin3bcos2b

(cos3b+cos5b+cosb)/(sin5b+sin3b+sinb)=(2cos3bcos2b+cos3b)/(2sin3bcos2b+sin3b)=

=cos3b(2cos2b+1)/sin3b(2cos2b+1)=cos3b/sin3b=ctg3b

Answer 2

To simplify the given expression, let's rewrite it as follows:

(cos 3b + cos 5b + cos b) / (sin b + sin 5b + sin 3b)

We can now focus on simplifying the numerator and denominator separately.

Numerator: Using the trigonometric identity cos(a) + cos(b) = 2cos((a+b)/2)cos((a-b)/2), we can simplify the numerator as follows:

cos 3b + cos 5b + cos b = 2cos(4b)cos(-b/2) + cos b = 2cos(4b)cos(b/2) + cos b = 2cos(4b)cos(b/2) + cos(b)

Denominator: Using the trigonometric identity sin(a) + sin(b) = 2sin((a+b)/2)cos((a-b)/2), we can simplify the denominator as follows:

sin b + sin 5b + sin 3b = 2sin(4b)cos(b/2) + sin b = 2sin(4b)cos(b/2) + sin(b)

Now, let's rewrite the simplified expression:

(2cos(4b)cos(b/2) + cos(b)) / (2sin(4b)cos(b/2) + sin(b))

Please note that further simplification may not be possible without additional information about the values of 'b'.