Cosecant
Cosecant of an angle in a right triangle
In a right triangle OAB, OA and AB are the legs, OB is the hypotenuse.The cosecant of an angle in a right triangle is equal to the ratio of the hypotenuse OB to
Designations: csc, csc α, csc x
The cosecant of an angle in a right triangle is equal to the ratio of the hypotenuse OB to opposite leg AB.
Designations: csc, csc α, csc xFor example, if the opposite leg is 2 and the hypotenuse is 4, then angle α will be equal to 30°, and cosecant of angle α will be 2/4=0.5
As you can see, if the angle tends to 0, then the leg AB will tend to 0 and cosecant of the angle in this limiting case will tend to infinity and in the end it will not be determined: OB/AB = OB/0 = X (±∞).
And in the same way, if the angle tends to 90 degrees, then the length of the opposite leg will tend to be equal to the length of the hypotenuse and cosecant of such an angle in the limiting case will be equal to 1: OB/AB = OB/OB = 1.
Cosecant of an arbitrary angle
The cosecant of an angle in a right triangle is the cosecant of an acute angle, a special case of the cosecant 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. And through the center of the coordinates, parallel to the radius, is the axis of cosecants. The cosecant axis is dynamic; it rotates along with the radius (hypotenuse) when the angle changes. The positive direction of the cosecant axis is codirectional with the radius. Cosecant of an arbitrary angle (acute, right, obtuse, straight, concave, complete) - the length of a segment with the origin at the coordinate center (in the center of the circle) - in zero of the cosecant axis, and ends at the point of intersection of the cosecant axis and the cotangent axis.
What is the relationship between the cosecant of an arbitrary angle and the cosecant in a right triangle? The radius OB, its projection OA onto the x-axis and the segment AB connecting the radius and the projection form such a right triangle. Triangle OA''B'' is similar to triangle OAB, which means the cosecant of the angle is also equal to the ratio of OB'' to OA''. OA'' is the radius of the circle and is equal to 1, which means the secant of the angle is equal to OB'' - a segment with the beginning at the coordinate center and ending at the intersection point of the continuation of the radius (cosecant axis) and the cotangent axis.
From this it follows that:
1. The cosecant values of angles from 0° to 180° are positive in the segment [1, ∞]
1. The cosecant values of angles from 0° to 180° are positive in the segment [1, ∞]
2. The cosecant values of angles from 180° to 360° are negative in the segment [-∞, -1]
3. And, summing up, the entire range of values [-∞; -1]⋃[1; ∞]
4. Angle values from -90° to 90° mirror the angle values from 90° to 180°
Cosecant values of some angles.
where N is the finger number starting from the little finger. The exception is the little finger (0°), for which the value is not defined
Mnemonic rule:
`csc α = 2/sqrt(N), `
| Finger number | 0 | 1 | 2 | 3 | 4 |
| Angle in degrees | 0° | 30° | 45° | 60° | 90° |
| Angle in radians | 0 | `pi/6` | `pi/4` | `pi/3` | `pi/2` |
| Value | ind | `2` | `sqrt(2)` | `(2sqrt(3))/3` | 1 |
Equation
The equation cscx = a means that you need to find angles x whose cosecant is equal to a. For example the equation
`text(csc) x = 1`
has solutions `x=pi/2+2pik, k∈Z`
That is, x is an infinite series of values `... -(7pi)/2, -(3pi)/2, pi/2, (5pi)/2, (9pi)/2 ...`
And the equation
`text(csc) x = 2`
has solutions `x=(-1)^k*pi/6+pik, k∈Z`
That is, x is an infinite series of values `... -(11pi)/6, -(7pi)/6, pi/6, (5pi)/6, (13pi)/6, (17pi)/6 ...`
Properties
Сompare properties
Function
`y = text(csc) x`
The cosecant function is inversely proportional to the sine function.
Function domain
`D(text(csc) x) = R - {pin, (n in Z)}`
The domain of definition of the cosecant is the entire numerical axis, with the exception of points at which the sine is equal to zero (and an uncertainty of the form 1/0 is obtained), i.e. except for points x = πn, n is any integer. On the circle, these are the boundaries between the I and IV quarters and between the II and III quarters.
Range
`E(text(csc) x) = [-infty, -1] uu [1, infty]`
The range of cosecant values is the entire numerical axis, with the exception of the interval from -1 to 1. The cosecant simply cannot become less than 1 in absolute value, since it is inversely proportional to the sine, which cannot be greater than 1 in absolute value.
Parity
`odd`
Like the sine, the cosecant is an odd function. When the angle value passes through 0, it also changes its sign. The function is symmetrical about the coordinate center.
Periodicity
`T = 2pi`
The trajectory of the cosecant function begins to repeat itself only after completing a full circle.
Zeros
`no`
Based on the range of values and the graph, it is clear that the function has no values between -1 and 1 and does not intersect the horizontal x-axis.
Constant sign intervals
`text(csc) x < 0,`
`x in (-pi+2pin, 2pin), n in Z`
`text(csc) x > 0,`
`x in (2pin, pi + 2pin), n in Z`
The cosecant function is negative in the III and IV quarters and positive in the I and II quarters of the circle.
Monotony intervals
`uarruarruarr x in (-pi+2pin, -pi/2 + 2pin]uu`
`uu[pi/2+2pin, pi+2pin), n in Z`
`darrdarrdarr x in [-pi/2+2pin, 2pin)uu`
`uu(2pin, pi/2+2pin], n in Z`
The cosecant function decreases in the I and IV quarters, and increases in the II and III quarters of the circle.
Extrema
`y_min = 1, x = pi/2+2pin, n in Z`
`y_max = -1, x = -pi/2+2pin, n in Z`
The cosecant function has local minima equal to 1 between the I and II quarters and local maxima equal to -1 between the III and IV quarters of the circle.
Inflection points
`no`
It is nowhere possible to draw a tangent to the cosecant function such that its local neighborhoods are on opposite sides of this tangent.
Formulas
`text(csc) alpha = 1/sin alpha`
`text(csc) alpha = (cot alpha)/cos alpha`
`text(csc) alpha = -sec(alpha+pi/2) = sec(alpha-pi/2)`
`-text(csc) alpha = text(csc)(-alpha)`
`(text(csc) alpha)^' = - cos alpha/sin^2 alpha`
`int (text(csc) alpha)dalpha = -ln |tan(alpha/2)| + C`