Use the sum and difference identities to determine the exact value of the following expression.sin(2π3−π4)
Question
Use the sum and difference identities to determine the exact value of the following expression.
sin(2π/3 - π/4)
Solution
Sure, let's use the sum and difference identities to solve the expression sin(2π/3 - π/4).
The difference identity for sine is: sin(a - b) = sin(a)cos(b) - cos(a)sin(b)
Let's plug in our values into this formula:
sin(2π/3 - π/4) = sin(2π/3)cos(π/4) - cos(2π/3)sin(π/4)
We know that sin(2π/3) = √3/2, cos(π/4) = √2/2, cos(2π/3) = -1/2, and sin(π/4) = √2/2.
Substitute these values into the equation:
= (√3/2 * √2/2) - (-1/2 * √2/2) = √6/4 + √2/4 = (√6 + √2) / 4
So, sin(2π/3 - π/4) = (√6 + √2) / 4.
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