Use standard identities to express sin(2θ+π/3) in terms of sinθ and cosθ
consider the identity Sin(A+B)=
then, look at the identties for double angles when A is 2Theta
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To express sin(2θ + π/3) in terms of sinθ and cosθ, we can use the standard trigonometric identity:
sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
In this case, A = 2θ and B = π/3. So we have:
sin(2θ + π/3) = sin(2θ)cos(π/3) + cos(2θ)sin(π/3)
Now, let's simplify the equation using the standard trigonometric identities for sin(π/3) and cos(π/3):
sin(π/3) = √3/2
cos(π/3) = 1/2
Substituting these values into the equation:
sin(2θ + π/3) = sin(2θ)(1/2) + cos(2θ)(√3/2)
Now, let's express sin(2θ) and cos(2θ) in terms of sinθ and cosθ, using the double-angle formulas:
sin(2θ) = 2sinθcosθ
cos(2θ) = cos^2θ - sin^2θ
Substituting these values into the equation:
sin(2θ + π/3) = (2sinθcosθ)(1/2) + (cos^2θ - sin^2θ)(√3/2)
Simplifying further:
sin(2θ + π/3) = sinθcosθ + (√3/2)(cos^2θ - sin^2θ)
And there you have it! You can express sin(2θ + π/3) in terms of sinθ and cosθ as sinθcosθ + (√3/2)(cos^2θ - sin^2θ).