Question

Tg(П/2+a),если ctga=4/3

Answer 1

Решение
Tg(П/2+a),если ctga=4/3
tg(π/2 + a) = - ctga = - 4/3

Answer 2

To solve this problem, we can use the identity:

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ctg(a) = 1 / tan(a)

Since ctg(a) = 4/3, we have:

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1 / tan(a) = 4/3

Multiplying both sides by tan(a), we get:

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tan(a) = 3/4

We know that:

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tg(П/2 + a) = -ctg(a)

So we have:

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tg(П/2 + a) = -ctg(a) = -(4/3)

Now we need to find the quadrant in which the angle (П/2 + a) lies. Since tg(П/2 + a) is negative, (П/2 + a) must lie in the second or fourth quadrant.

Since tan(a) = 3/4 is positive, a must lie in the first or third quadrant. Therefore, (П/2 + a) lies in the fourth quadrant, where both tg and ctg are negative.

Using the fact that tg(П/2 + a) = -(4/3) and (П/2 + a) lies in the fourth quadrant, we can use the reference angle of a right triangle to find the value of tg(П/2 + a):

Let's consider a right triangle with an angle x in the fourth quadrant. The reference angle of x is the acute angle formed by the terminal side of x and the x-axis. Let's call this reference angle y.

Since tan(y) = 3/4, we can label the opposite side of y as 3 and the adjacent side as 4. Using the Pythagorean theorem, we can find the hypotenuse:

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hypotenuse^2 = 3^2 + 4^2 = 25
hypotenuse = sqrt(25) = 5

Now we can find the value of tg(y):

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tg(y) = opposite / adjacent = 3/4

Since (П/2 + a) is the supplementary angle of y, we have:

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tg(П/2 + a) = -tg(y) = -(3/4) = -0.75

Therefore, tg(П/2 + a) = -0.75 when ctg(a) = 4/3 and (П/2 + a) lies in the fourth quadrant.