The cotangent function ‘or’ cot theta is one of the trigonometric functions apart from sine, cosine, tangent, secant, and cosecant. The cotangent function in right-angle triangle trigonometry is defined as the ratio of the adjacent side to lớn the opposite side.

Bạn đang xem: Cotangent

The mathematical denotation of the cotangent is,

(cot( heta) = frac extAdjacent Side extOpposite Side)

Cot theta can also be calculated from the ratio of the cosine of the angle to the sine of the angle.

(cot( heta) = fraccos( heta)sin( heta))

The derivative of (cot( heta)) in calculus is (-cosec^2( heta)) and the integral of it is (ln|sin( heta)|). The reciprocal of (cot( heta)) is ( an( heta)).

Graph of the cot theta function

Below is a table of cotangent values for different degrees và radians.

 Radians Degree Cotangent Value 0 0° (infty) (fracpi6) 30° (frac3sqrt3) (fracpi4) 45° 1 (fracpi3) 60° (frac1sqrt3) (fracpi2) 90° 0 (pi) 180° (infty) (frac3pi2) 270° 0 (2pi) 360° (infty)

## Important Cot Theta Formula

Some important properties of the cotangent function and cot theta formula are:

(cot(-x) = -cot(x))(cot(90°-x) = an(x))(cot(x + pi) = cot(x))(cot(pi – x) = -cot(x))(cot^2(x) = cosec^2(x) – 1)(cot(x + y) = fraccot(x) + cot(y)cot(x) * cot(y) – 1)(cot(x – y) = fraccot(y) – cot(x)cot(x) * cot(y) + 1)

## Solved Examples

Question 1. If (sin(x) = frac45), calculate the value of (cot(x)).

Solution. Using trigonometric identity,

(cos^2(x) = 1 – sin^2(x) = 1 – frac1625 = frac925)

(cos(x) = frac35)

Now,

( an(x) = fracsin(x)cos(x) = fracfrac45frac35)

( an(x) = frac43)

As we know,

(cot(x) = frac1 an(x))

(∴ cot(x) = frac34)

Question 2. If (sec(x) = frac58), calculate the value of ( an(x)).

Solution. Using trigonometric identity,

( an^2(x) = sec^2(x) – 1 = (frac58)^2 – 1 = frac2564 – 1 = -frac3964)

(∴ an(x) = sqrt-frac3964)

Question 3. John is standing on the ground và looking at the đứng đầu of a tower with an angle of elevation of 60°. The distance between John và the tower is 15 feet. Calculate the height of the tower.

Solution. As we know,

( an( heta) = frac extOpposite extAdjacent)

( an(60°) = frac extOpposite ext15)

(sqrt3 = frac extOpposite ext15)

( extOpposite = 15 sqrt3)

∴ The height of the tower is (15 sqrt3) feet.

## FAQs

Explain how cot(-x) = -cot(x).

As we know, (cot(x) = fracsin(x)cos(x)). The angle (-x) lies in the 4th quadrant of a graph, và sine is negative in this quadrant và cos is positive. So, the ratio is negative. Hence, this shows that cot(-x) = -cot(x).

How is cot theta defined?

Cot theta of a right-angled triangle is equal to lớn the ratio of the length of the opposite side khổng lồ the length of the adjacent side. It is also equal to lớn the ratio of the cosine of the angle và the sine of the angle.

Xem thêm: Cách Viết Phương Trình Hóa Học, Bài Tập Viết Phương Trình Hóa Học

In which quadrants is the cot function positive and in which quadrants is it negative?

It can be observed from the above graph that cot(x) is positive in the 1st and 3rd quadrants & negative in the 2nd & 4th quadrants.