Question

В каких четвертях отрицательно выражение ctgx*sinx?

Answer 1

Ctgx*sinx=(cosx/sinx)*sinx=cosx
cosx отрицателени во II и в III  координатных четвертях
Следовательно исходное выражение принимает отрицательные значения
во II и в III координатных четвертях.

Answer 2

Expression ctgx*sinx in Negative Quarters

The expression ctgx*sinx can be negative in certain quarters. Let's explore the conditions under which this expression can be negative.

In trigonometry, the cotangent function (ctgx) is the reciprocal of the tangent function (tgx). The cotangent of an angle in a right-angled triangle is the ratio of the length of the adjacent side to the length of the opposite side. The sine function (sinx) represents the ratio of the length of the side opposite the given angle to the length of the hypotenuse.

When considering the signs of trigonometric functions in different quarters, it's important to note that: - In the first quarter (0° to 90°), all trigonometric functions are positive. - In the second quarter (90° to 180°), only the sine function is positive. - In the third quarter (180° to 270°), only the tangent function is positive. - In the fourth quarter (270° to 360°), only the cotangent function is positive.

Given the above information, we can determine the conditions under which the expression ctgx*sinx is negative.

Conditions for ctgx*sinx to be Negative: 1. The expression ctgx*sinx can be negative in the second and fourth quarters. 2. In the second quarter, sinx is positive, and ctgx is negative, resulting in a negative product. 3. In the fourth quarter, sinx is negative, and ctgx is positive, resulting in a negative product.

Therefore, the expression ctgx*sinx can be negative in the second and fourth quarters.