Question

(tg(a)+ctg(a))/tg(a)-ctg(a))=...Если tg(a)=2/3​

Answer 1

Ответ:

$-2,6$

Объяснение:

*** При решении использованы формулы:

$tga=\frac{sina}{cosa},\;\;\;\;\;ctga=\frac{cosa}{sina}\\\\sin^2a+cos^2a=1\\cos^2a-sin^2a=cos2a\\1+tg^2a=\frac{1}{cos^2a}$

Решение:

$tga=2/3\\\\\frac{tga+ctga}{tga-ctga}=\frac{\frac{sina}{cosa}+\frac{cosa}{sina}}{\frac{sina}{cosa}-\frac{cosa}{sina}}=\frac{\frac{sin^2a+cos^2a}{sinacosa}}{\frac{sin^2a-cos^2a}{sinacosa}}=\frac{1}{sin^2a-cos^2a}=\frac{1}{1-cos^2a-cos^2a}=\\\\\\=\frac{1}{1-2cos^2a}=\frac{1}{1-\frac{2}{1+tg^2a}}=\frac{1}{\frac{1+tg^2a-2}{1+tg^2a}}=\frac{1+tg^2a}{tg^2a-1}\\\\\\\frac{1+tg^2a}{tg^2a-1}=\frac{1+(2/3)^2}{(2/3)^2-1}=\frac{1+(4/9)}{(4/9)-1}=\frac{13/9}{-5/9}=-\frac{13}{5}=-2,6$

Answer 2

To find the value of the expression when tg(a) = 2/3, we first need to determine the value of ctg(a) (cotangent).

Recall that cotangent (ctg) is the reciprocal of tangent (tg). Therefore, if tg(a) = 2/3, then ctg(a) = 1 / tg(a) = 1 / (2/3) = 3/2.

Now, let's substitute tg(a) = 2/3 and ctg(a) = 3/2 into the expression:

(tg(a) + ctg(a)) / (tg(a) - ctg(a)) = (2/3 + 3/2) / (2/3 - 3/2)

To perform the operations with fractions, let's find a common denominator, which is 6:

(2/3 + 3/2) = (4/6 + 9/6) = 13/6 (2/3 - 3/2) = (4/6 - 9/6) = -5/6

Now, the expression becomes:

(13/6) / (-5/6)

When dividing fractions, we multiply by the reciprocal of the denominator:

(13/6) * (-6/5) = -13/5

So, when tg(a) = 2/3, the expression (tg(a) + ctg(a)) / (tg(a) - ctg(a)) evaluates to -13/5.