Question

Вычислите: cos2a-cos6a, если cosa=1/√3

Answer 1

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Ответ дробный получился.

Answer 2

To compute the value of cos(2a) - cos(6a), given that cos(a) = 1/√3, we can use trigonometric identities and substitution to simplify the expression.

First, let's find the value of cos(2a). Using the double-angle formula for cosine, we have:

cos(2a) = 2cos^2(a) - 1

Substituting the value of cos(a) = 1/√3, we get:

cos(2a) = 2(1/√3)^2 - 1 = 2/3 - 1 = -1/3

Next, let's find the value of cos(6a). Using the triple-angle formula for cosine, we have:

cos(6a) = 4cos^3(a) - 3cos(a)

Substituting the value of cos(a) = 1/√3, we get:

cos(6a) = 4(1/√3)^3 - 3(1/√3) = 4/27√3 - 3/√3 = (4 - 27) / 27√3 = -23/27√3

Now we can substitute both values into the expression cos(2a) - cos(6a):

cos(2a) - cos(6a) = (-1/3) - (-23/27√3) = -1/3 + 23/27√3

To simplify this further, we need to rationalize the denominator by multiplying the numerator and denominator by the conjugate of 27√3, which is 27√3:

cos(2a) - cos(6a) = (-1/3) + 23/27√3 * (27√3/27√3) = -1/3 + (23 * 27√3) / (27 * 3) = -1/3 + (23 * √3) / 3

Finally, we can combine the terms:

cos(2a) - cos(6a) = (-1 + 23√3) / 3

Therefore, the value of cos(2a) - cos(6a), given cos(a) = 1/√3, is (-1 + 23√3) / 3.