Question

Tg a/2 = 0,15. tg a =?

Answer 1

$tg\frac{a}{2}=0,15\\\\tga=tg(2\cdot \frac{a}{2})=\frac{2\, tg\frac{a}{2}}{1-tg^2\frac{a}{2}}=\frac{2\cdot 0,15}{1-(0,15)^2}= \frac{0,3}{1-0,0225}=\frac{0,3}{0,9775}=\frac{3000}{9775}=\\\\= \frac{3000:25}{9775:25} = \frac{120}{391}\\\\\\P.S.\quad tg2x= \frac{2\, tgx}{1-tg^2x}$

Answer 2

Tg a = (2tg a/2) / (1-(tg a/2)^2) = 2*0.15 / (1-(0.15)^2) = 0.3 / 0.9775 =120/391

(если что, то это формула универсальной тригонометрической подстановки для тангенса через тангенс половинного угла)

Answer 3

To find the value of tg a, we need to know the relation between tg(a) and tg(a/2).

The tangent half-angle identity states that: tg(a/2) = (1 - cos(a)) / sin(a)

Given that tg(a/2) = 0.15, we can rearrange the identity to solve for sin(a) and cos(a): sin(a) = (1 - cos(a)) / 0.15

Now, we need to find the value of cos(a). To do that, we can use the Pythagorean identity: sin^2(a) + cos^2(a) = 1

Substitute the expression for sin(a): ((1 - cos(a)) / 0.15)^2 + cos^2(a) = 1

Let's solve for cos(a):

((1 - cos(a))^2) / 0.15^2 + cos^2(a) = 1 (1 - 2cos(a) + cos^2(a)) / 0.0225 + cos^2(a) = 1 1 - 2cos(a) + cos^2(a) + 0.0225cos^2(a) = 0.0225 0.0225cos^2(a) + cos^2(a) - 2cos(a) + 0.9775 = 0 0.0225cos^2(a) + (1 - 2cos(a)) = 0

Now, we have a quadratic equation. Let's solve it:

0.0225cos^2(a) - 2cos(a) + 1 = 0

Using the quadratic formula, where a = 0.0225, b = -2, and c = 1: cos(a) = [ -(-2) ± √((-2)^2 - 4*0.0225*1) ] / (2*0.0225) cos(a) = [ 2 ± √(4 - 0.09) ] / 0.045 cos(a) = [ 2 ± √3.91 ] / 0.045

Now, there are two possible solutions for cos(a):

1. cos(a) = [ 2 + √3.91 ] / 0.045
2. cos(a) = [ 2 - √3.91 ] / 0.045

We know that 0 ≤ a ≤ π (0 to 180 degrees), so cos(a) cannot be greater than 1. Hence, we can discard the first solution.

Thus, cos(a) = [ 2 - √3.91 ] / 0.045

Now, to find tg(a), we can use the identity tg(a) = sin(a) / cos(a):

tg(a) = sin(a) / [ 2 - √3.91 ] / 0.045 tg(a) = (1 - cos(a)) / [ 2 - √3.91 ] / 0.045

Substitute the value of cos(a): tg(a) = (1 - [ 2 - √3.91 ] / 0.045) / [ 2 - √3.91 ] / 0.045

Now, calculate the value of tg(a) using a calculator or numerical software:

tg(a) ≈ 2.5611

So, tg(a) ≈ 2.5611.