Question

Вычислить: sin7П/12 - cosП/12

Answer 1

sin7П/12 - cosП/12=                 разложим первый аргумент
sin(6П/12 + П/12) - cosП/12=   упростим немного
sin(П/2 + П/12) - cosП/12=       заменим на кофункцию
cosП/12- cosП/12=                   выполним вычитание
0.

Answer 2

To calculate the value of the expression sin(7π/12) - cos(π/12), we'll need to know the exact values of sine and cosine at specific angles.

Here are the values of sine and cosine at some common angles:

sin(π/6) = 1/2 cos(π/6) = √3/2 sin(π/4) = √2/2 cos(π/4) = √2/2 sin(π/3) = √3/2 cos(π/3) = 1/2

We can use these values to find sin(7π/12) and cos(π/12) since 7π/12 and π/12 are not common angles.

We can use the following trigonometric identities:

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

Let's calculate sin(7π/12) and cos(π/12):

sin(7π/12) = sin(π/3 + π/4) = sin(π/3) * cos(π/4) + cos(π/3) * sin(π/4) sin(7π/12) = (√3/2) * (√2/2) + (1/2) * (√2/2) sin(7π/12) = (√6 + √2) / 4

cos(π/12) = cos(π/3 - π/4) = cos(π/3) * cos(π/4) + sin(π/3) * sin(π/4) cos(π/12) = (1/2) * (√2/2) + (√3/2) * (√2/2) cos(π/12) = (√2 + √6) / 4

Now, let's calculate the expression sin(7π/12) - cos(π/12):

sin(7π/12) - cos(π/12) = (√6 + √2) / 4 - (√2 + √6) / 4 sin(7π/12) - cos(π/12) = (√6 + √2 - √2 - √6) / 4 sin(7π/12) - cos(π/12) = 0

So, the value of the expression sin(7π/12) - cos(π/12) is 0.