Question

Вычислите:2-4sin^2 5pi/12

Answer 1

Ответ в кружочке. Реш-е во вложении.

Answer 2

To calculate the expression, we need to first find the value of sin^2(5π/12) and then proceed with the rest of the calculation.

Step 1: Calculate sin^2(5π/12) Using the trigonometric identity: sin^2θ = (1 - cos(2θ))/2

We have θ = 5π/12, so: sin^2(5π/12) = (1 - cos(2 * 5π/12))/2

Step 2: Calculate cos(2 * 5π/12) Using the trigonometric identity: cos(2θ) = cos^2θ - sin^2θ

Let's find cos(5π/12) first. You can use the following trigonometric half-angle formula to calculate it:

cos(θ/2) = ±√((1 + cosθ)/2)

In this case, θ = 5π/6, so: cos(5π/12) = ±√((1 + cos(5π/6))/2) cos(5π/6) = ±√((1 + (√3/2))/2) cos(5π/6) = ±√((2 + √3)/4)

Since 5π/12 lies in the second quadrant (180 degrees to 270 degrees) where cosine is negative, we take the negative root:

cos(5π/12) = -√((2 + √3)/4)

Now, we can calculate cos(2 * 5π/12):

cos(2 * 5π/12) = cos(5π/6) = -√((2 + √3)/4)

Step 3: Substitute cos(2 * 5π/12) into the expression for sin^2(5π/12):

sin^2(5π/12) = (1 - cos(2 * 5π/12))/2 sin^2(5π/12) = (1 - (-√((2 + √3)/4)))/2 sin^2(5π/12) = (1 + √((2 + √3)/4))/2 sin^2(5π/12) = (2 + √(2 + √3))/4

Step 4: Calculate the final result:

2 - 4sin^2(5π/12) = 2 - 4 * (2 + √(2 + √3))/4 2 - 4sin^2(5π/12) = 2 - (2 + √(2 + √3)) 2 - 4sin^2(5π/12) = 2 - 2 - √(2 + √3) 2 - 4sin^2(5π/12) = -√(2 + √3)

So, the final result is -√(2 + √3).