Question

Sina= -4/5 , 180°<a <270°. надо найти cosa и ctga

Answer 1

Дано: sinα =  $- \frac{4}{5}$
180°<α<270° (угол в III четверти)
Найти: cosα, ctgα
Решение:
1) Основное тригонометрическое тождество: sin²α + cos²α = 1, тогда
cos²α = 1 - sin²α = 1 - (- $\frac{4}{5}$)² = 1 - $\frac{16}{25}$ =  $\frac{25-16}{25}$ =  $\frac{9}{25}$
Учитывая, что угол находится в III четверти, cosα<0, получаем
cosα = -√ $\frac{9}{25}$ =  $- \frac{3}{5}$

2) ctgα = cosα/sinα
Получаем
ctgα =  $- \frac{3}{5} : - \frac{4}{5} = \frac{3}{5} * \frac{5}{4} = \frac{3}{4}$ 

Ответ: cosα =  $- \frac{3}{5}$
ctgα =  $\frac{3}{4}$

Answer 2

To find the values of cos(a) and cot(a), we first need to determine the quadrant in which the angle "a" lies. The given condition states that 180° < a < 270°, which means "a" lies in the third quadrant (Q3).

In the third quadrant, cosine is negative, while cotangent is positive.

1. Finding cos(a): Since the given value "Sina" is -4/5, we can use the Pythagorean identity to find cos(a): cos(a) = ±√(1 - sin^2(a))

Given sin(a) = -4/5, we have: cos(a) = ±√(1 - (-4/5)^2) = ±√(1 - 16/25) = ±√(9/25) = ±3/5

Since "a" lies in the third quadrant (Q3), cos(a) will be negative: cos(a) = -3/5

2. Finding cot(a): cot(a) can be calculated as the reciprocal of the tangent function: cot(a) = 1/tan(a)

In the third quadrant, the tangent function is positive, so we can use the given value of sin(a) to find tan(a): sin(a) = -4/5 tan(a) = sin(a)/cos(a) = (-4/5) / (-3/5) = 4/3

Now we can find cot(a): cot(a) = 1/tan(a) = 1 / (4/3) = 3/4

Therefore, in the given scenario where -4/5 and 180° < a < 270°, we have: cos(a) = -3/5 cot(a) = 3/4