Question

-2tg(-П+у)-2tg(-y),если tgy=0,5

Answer 1

-2tg(-П+у)-2tg(-y) = 2tg(П-у)+2tg(y) = -2tg(у)+2tg(y)  = 0

Answer 2

First, we can use the trigonometric identity tan^2(x) + 1 = sec^2(x) to rewrite tgy in terms of cosy:

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tgy = sin(y) / cos(y)
    = √(1 - cos^2(y)) / cos(y)
    = 1 / √(sec^2(y) - 1)
    = 1 / √(1/cos^2(y) - 1)
    = 1 / √(1 - cos^2(y)/cos^2(y))
    = 1 / √(sin^2(y)/cos^2(y))
    = cos(y)/sin(y)

Since we know that tgy = 0.5, we can solve for cos(y)/sin(y):

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tgy = cos(y)/sin(y)
0.5 = cos(y)/sin(y)
cos(y) = 0.5 sin(y)

We can use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to solve for sin(y):

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sin^2(y) + cos^2(y) = 1
sin^2(y) + (0.5 sin(y))^2 = 1
1.25 sin^2(y) = 1
sin^2(y) = 1/1.25
sin(y) = ±0.8944

Since y must be between -π/2 and π/2 (so that tgy is positive), we can take the positive value for sin(y):

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sin(y) = 0.8944
cos(y) = 0.5 sin(y) = 0.4472

Now we can substitute cos(y) and sin(y) into the expression -2tg(-π+у)-2tg(-y):

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-2tan(-π+y) - 2tan(-y)
= 2tan(y-π) + 2tan(y)
= 2(-tan(π-y)) + 2tan(y)  (using the identity tan(-x) = -tan(x))
= 2(tan(y) - tan(π-y))
= 2(tan(y) + tan(y))
= 4tan(y)
= 4sin(y)/cos(y)
= 4(0.8944)/0.4472
= 8

Therefore, -2tg(-π+у)-2tg(-y) = 8 when tgy = 0.5.