Cotangent
Cotangent of an angle in a right triangle
In a right triangle OAB, OA and AB are the legs, OB is the hypotenuse.The cotangent of an angle in a right triangle is equal to the ratio of the
Designations: cot, cot α, cot x
The cotangent of an angle in a right triangle is equal to the ratio of the adjacent leg OA к opposite leg AB .
Designations: cot, cot α, cot xFor example, if the adjacent leg is equal to 1, and the opposite leg is also equal to 1, then the angle α will be equal to 45°, and the cotangent of the angle α will be 1 / 1 = 1
As you can see, if the angle tends to 0, then the length of the adjacent leg OA will tend to equal the length of the hypotenuse, the length of the opposite leg AB will tend to 0 and angle cotangent in this limiting case will tend to infinity and in the end it will not be determined: OA/AB = OA/0 = X (±∞).
And in the same way, if the angle tends to 90 degrees, then the length of the adjacent leg OA will tend to 0, the length of the opposite leg AB will tend to be equal with the length of the hypotenuse and cotangent of the angle in this limiting case will tend to 0: OA/AB = 0/AB = 0.
Cotangent of an arbitrary angle
The cotangent of an angle in a right triangle is the cotangent of an acute angle, a special case of the cotangent of an arbitrary angle
Let us draw an additional horizontal axis parallel to the abscissa axis through the point (0; 1) - the axis of cotangents. Cotangent of an arbitrary angle (acute, straight, obtuse, straight, concave, full) - abscissa of a point (horizontal coordinate x) on the cotangent axis obtained by rotating the point (1; 0) around the origin by an angle counterclockwise arrows on the unit circle and the continuation of the radius segment until it intersects with the cotangent axis.
What is the relationship between the cotangent of an arbitrary angle and the cotangent in a right triangle? The cotangent of an angle in a right triangle OAB is equal to the ratio of the adjacent side OA to the opposite side AB. Triangles OAB and OA''B'' are similar, which means the cotangent of the angle is also equal to the ratio of A''B'' to OA''. OA'' is the radius of the circle and is equal to 1. Therefore, the cotangent of the angle is equal to A''B'' or the abscissa of the intersection point of the continuation of the radius and the axis of the cotangents.
It follows from this that:
1. The values of the cotangent of angles from 0° to 90° and from 180° to 270° are positive in the segment [0, +∞]
1. The values of the cotangent of angles from 0° to 90° and from 180° to 270° are positive in the segment [0, +∞]
2. Values of cotangent angles from 90° to 180° and from 270° to 360° are negative in the segment [-∞, 0]
3. And, summing up, the entire range of values [-∞; ∞]
4. Values for angles between 0° and 180° are not defined
5. Angle values from 0° to 180° centrally symmetrically repeat angle values from 90° to 270°
Cotangent values of some angles.
where N is the number of the finger starting from the thumb. The exception is the little finger (0°), for which the value is not defined
Mnemonic rule:
`cot α = sqrt(3)^N/3, `
| Finger number | 0 | 1 | 2 | 3 | 4 |
| Angle in degrees | 90° | 60° | 45° | 30° | 0° |
| Angle in radians | `pi/2` | `pi/3` | `pi/4` | `pi/6` | 0 |
| Value | 0 | `sqrt(3)/3` | `1` | `sqrt(3)` | ind |
Equation
The equation cotx = a means that you need to find angles x whose cotangent is equal to a. For example the equation
`cot x = 1`
has solutions `x=pi/4+pik, k∈Z`
То есть x это бесконечный ряд значений `... -(7pi)/4, -(3pi)/4, (pi)/4, (5pi)/4, (9pi)/4, ...`
And the equation
`cot x = 0`
has solutions `x=pi/2 + pik, k∈Z`
That is, x is an infinite series of values `... -(3pi)/2, -pi/2, pi/2, (3pi)/2, (5pi)/2, ...`
Properties
Сompare properties
Function
`y = cot x`
The cotangent function is inversely proportional to the tangent function.
Function domain
`D(cot x) = R - {pin (n in Z)}`
The domain of definition of the cotangent is the entire numerical axis, with the exception of points at which the opposite leg (or sine of the angle) is equal to zero and an uncertainty of the form 1/0 is obtained. On the circle, these are the boundaries between the I and IV quarters and between the II and III quarters.
Range
`E(cot x) = R`
The range of cotangent values is the entire numerical axis.
Parity
`odd`
When the angle value passes through 0, the cotangent function of this angle also changes its sign. The function is symmetrical about the coordinate center.
Periodicity
`T = pi`
The trajectory of the cotangent function is repeated every half circle.
Zeros
`cot x = 0, x=pi/2 + pin, n in Z`
The cotangent function has zeros at points between the I and II quarters and between the III and IV quarters of the circle.
Constant sign intervals
`cot x > 0, x in (pin, pi/2 + pin), n in Z`
`cot x < 0, x in (-pi/2 + pin, pin), n in Z`
The cotangent function is negative in the II and IV quarters and positive in the I and III quarters of the circle.
Monotony intervals
`darrdarrdarr x in D`
The cotangent function decreases over the entire domain of its definition. That is, it always decreases.
Extrema
`no`
The cotangent function has no minimums or maximums, since it always decreases.
Inflection points
`y=0, x = pi/2 + pin, n in Z`
The cotangent function has inflection points between the I and II quarters and between the III and III quarters of the circle with a value of 0.
Formulas
`cot alpha = cos alpha / sin alpha`
`cot alpha = 1/(tan alpha)`
`cot alpha = - tan(alpha + pi/2) = - tan(alpha - pi/2)`
`cot alpha = sqrt(1/sin^2alpha - 1)`
`-cot alpha = cot(-alpha)`
`(cot alpha)^' =-1/sin^2 alpha`
`int cot(alpha)dalpha = -ln |sin alpha| + C`