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To simplify the expression 5 cos(π/3) cos(2π/3) sin(π-π/3), we can use the trigonometric identities.
1. Recall the double angle identity for cosine: cos(2θ) = 2cos^2(θ) - 1.
2. Also, remember the difference of angles identity for sine: sin(θ - φ) = sin(θ)cos(φ) - cos(θ)sin(φ).
Now let's simplify the expression step by step:
1. cos(2π/3) = 2cos^2(π/3) - 1
(Using the double angle identity)
2. cos(2π/3) = 2(1/2)^2 - 1
(Substituting cos(π/3) = 1/2)
3. cos(2π/3) = 1/2 - 1
(Simplifying)
4. cos(2π/3) = -1/2
Now let's simplify the remaining part of the expression:
5. sin(π - π/3) = sin(2π/3)
(Using the difference of angles identity)
6. sin(2π/3) = sin(π/3)
(Sine function is periodic with period 2π)
7. sin(π/3) = √3/2
(Using the value of sin(π/3))
Finally, we can substitute the values back into the original expression:
5 cos(π/3) cos(2π/3) sin(π-π/3) = 5 * (1/2) * (-1/2) * (√3/2)
= -5/4 * √3/2
= -5√3/8
So, the simplified expression is -5√3/8.
1. Recall the double angle identity for cosine: cos(2θ) = 2cos^2(θ) - 1.
2. Also, remember the difference of angles identity for sine: sin(θ - φ) = sin(θ)cos(φ) - cos(θ)sin(φ).
Now let's simplify the expression step by step:
1. cos(2π/3) = 2cos^2(π/3) - 1
(Using the double angle identity)
2. cos(2π/3) = 2(1/2)^2 - 1
(Substituting cos(π/3) = 1/2)
3. cos(2π/3) = 1/2 - 1
(Simplifying)
4. cos(2π/3) = -1/2
Now let's simplify the remaining part of the expression:
5. sin(π - π/3) = sin(2π/3)
(Using the difference of angles identity)
6. sin(2π/3) = sin(π/3)
(Sine function is periodic with period 2π)
7. sin(π/3) = √3/2
(Using the value of sin(π/3))
Finally, we can substitute the values back into the original expression:
5 cos(π/3) cos(2π/3) sin(π-π/3) = 5 * (1/2) * (-1/2) * (√3/2)
= -5/4 * √3/2
= -5√3/8
So, the simplified expression is -5√3/8.
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29 января 2024 05:12
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