Question

Как выразить tg 22 градусов 30 минут? ctg 22 градусов 30 минут?

Answer 1

tg (22°30') > 0 ;  сtg (22°30') > 0 - первая четверть

$tg (22\textdegree 30')=\sqrt{\dfrac{1-\cos (45\textdegree)}{1+\cos (45\textdegree)}}=\sqrt\dfrac{1-\frac{\sqrt2}2}{1+\frac{\sqrt2}2}}=\\\\\\=\sqrt{\dfrac{2-\sqrt2}{2+\sqrt2}}=\sqrt{\dfrac{(2-\sqrt2)^2}{(2+\sqrt2)\cdot (2-\sqrt2)}}=\\\\\\=\sqrt{\dfrac{4-4\sqrt2+2}{2}}\boldsymbol{=\sqrt{3-2\sqrt2}}$

$ctg (22\textdegree 30')=\sqrt{\dfrac{1+\cos (45\textdegree)}{1-\cos (45\textdegree)}}=\sqrt\dfrac{1+\frac{\sqrt2}2}{1-\frac{\sqrt2}2}}=\\\\\\=\sqrt{\dfrac{2+\sqrt2}{2-\sqrt2}}=\sqrt{\dfrac{(2+\sqrt2)^2}{(2-\sqrt2)\cdot (2+\sqrt2)}}=\\\\\\=\sqrt{\dfrac{4+4\sqrt2+2}{2}}\boldsymbol{=\sqrt{3+2\sqrt2}}$

Проверка :

$tg (22\textdegree 30')\cdot ctg (22\textdegree 30')=\\\\=\sqrt{3-2\sqrt2}\cdot \sqrt{3+2\sqrt2}=3^2-(2\sqrt 2)^2=1$

======================================

Использованы формулы

$tg^2\dfrac {\alpha }2=\dfrac{1-\cos \alpha }{1+ \cos \alpha }\\\\ctg^2\dfrac {\alpha }2=\dfrac{1+\cos \alpha }{1- \cos \alpha }\\\\a^2-b^2=(a-b)(a+b)$

Answer 2

Expressing tg 22 degrees 30 minutes

To express tg 22 degrees 30 minutes, we need to convert the angle to decimal form. Here's how we can do it:

1. Convert the minutes to decimal form by dividing them by 60. In this case, 30 minutes divided by 60 equals 0.5. 2. Add the decimal form of the minutes to the degrees. In this case, 22 degrees plus 0.5 equals 22.5 degrees.

Therefore, tg 22 degrees 30 minutes can be expressed as tg 22.5 degrees.

Expressing ctg 22 degrees 30 minutes

To express ctg 22 degrees 30 minutes, we can use the reciprocal relationship between tangent (tg) and cotangent (ctg). The reciprocal of tangent is cotangent.

Therefore, ctg 22 degrees 30 minutes is the reciprocal of tg 22 degrees 30 minutes. Since we already determined that tg 22 degrees 30 minutes is equivalent to tg 22.5 degrees, we can express ctg 22 degrees 30 minutes as ctg 22.5 degrees.

Please note that the above explanations are based on mathematical principles and conversions.