Question

(sin5п/12+sin п/12)^2

Answer 1

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Answer 2

To square the expression (sin(5π/12) + sin(π/12))^2, you can use the binomial formula:

(a + b)^2 = a^2 + 2ab + b^2

In this case, let a = sin(5π/12) and b = sin(π/12):

(sin(5π/12) + sin(π/12))^2 = sin^2(5π/12) + 2sin(5π/12)sin(π/12) + sin^2(π/12)

Now, you can calculate each part of the expression:

1. sin^2(5π/12): Use the double angle formula for sine: sin(2θ) = 2sin(θ)cos(θ)

   So, sin(5π/12) = sin(π/3) = √3/2 sin^2(5π/12) = (√3/2)^2 = 3/4

2. 2sin(5π/12)sin(π/12): Use the product-to-sum identity: 2sin(θ)sin(φ) = cos(θ - φ) - cos(θ + φ)

   2sin(5π/12)sin(π/12) = cos(5π/12 - π/12) - cos(5π/12 + π/12) = cos(4π/12) - cos(6π/12) = cos(π/3) - cos(π/2) = (1/2) - 0 = 1/2

3. sin^2(π/12): Using the sine of a smaller angle (π/12): sin(π/12) = √[(1 - cos(π/6))/2] = √[(1 - √3/2)/2]

Now, plug these values back into the original expression:

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

Now, you can add the fractions:

= (3/4 + 1/2 + 1/2 - √3/4)^2 = (2 - √3/4)^2

To square this expression, simply square both the whole number part and the fraction part:

= (4 - √3)^2 = 16 - 8√3 + 3 = 19 - 8√3

So, (sin(5π/12) + sin(π/12))^2 = 19 - 8√3.