Question

Sin pi/12*cos pi/12*cos pi/6

Answer 1

Вот тут........................

Answer 2

To evaluate the expression sin(pi/12) * cos(pi/12) * cos(pi/6), we can break it down step by step.

Step 1: Evaluate sin(pi/12) The sine function (sin) returns the ratio of the length of the side opposite to the angle to the length of the hypotenuse in a right triangle. In this case, we need to find the value of sin(pi/12).

The angle pi/12 is equivalent to 15 degrees. To evaluate sin(15 degrees), we can use the trigonometric identity:

sin(a + b) = sin(a) * cos(b) + cos(a) * sin(b)

In this case, we have a = 0 and b = 15 degrees.

sin(0 + 15) = sin(0) * cos(15) + cos(0) * sin(15) sin(15) = 0 * cos(15) + 1 * sin(15) sin(15) = sin(15)

Therefore, sin(pi/12) is equal to sin(15 degrees).

Step 2: Evaluate cos(pi/12) The cosine function (cos) returns the ratio of the length of the adjacent side to the angle to the length of the hypotenuse in a right triangle. In this case, we need to find the value of cos(pi/12).

Using the same trigonometric identity as before:

cos(a + b) = cos(a) * cos(b) - sin(a) * sin(b)

In this case, we have a = 0 and b = 15 degrees.

cos(0 + 15) = cos(0) * cos(15) - sin(0) * sin(15) cos(15) = 1 * cos(15) - 0 * sin(15) cos(15) = cos(15)

Therefore, cos(pi/12) is equal to cos(15 degrees).

Step 3: Evaluate cos(pi/6) The angle pi/6 is equivalent to 30 degrees. To evaluate cos(30 degrees), we can use the trigonometric identity:

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

In this case, we have a = 30 degrees.

cos(30) = sqrt(1 - sin^2(30)) cos(30) = sqrt(1 - (1/2)^2) cos(30) = sqrt(1 - 1/4) cos(30) = sqrt(3/4) cos(30) = sqrt(3)/2

Now that we have the values for sin(pi/12), cos(pi/12), and cos(pi/6), we can substitute them back into the original expression:

sin(pi/12) * cos(pi/12) * cos(pi/6) = sin(15 degrees) * cos(15 degrees) * sqrt(3)/2

Since sin(15 degrees) = cos(15 degrees), we can simplify the expression further:

sin(pi/12) * cos(pi/12) * cos(pi/6) = (sin(15 degrees) * cos(15 degrees))^2 * sqrt(3)/2

Finally, we can square sin(15 degrees) * cos(15 degrees) and multiply by sqrt(3)/2 to get the result:

(sin(15 degrees) * cos(15 degrees))^2 * sqrt(3)/2 = (sin(15 degrees))^2 * (cos(15 degrees))^2 * sqrt(3)/2

I hope this explanation helps! Let me know if you have any further questions.