Secant
Secant of the angle in a right triangle
In a right triangle OAB, OA and AB are the legs, OB is the hypotenuse.The secant of an angle in a right triangle is equal to the ratio of the hypotenuse OB to
Designations: sec, sec α, sec x
The secant of an angle in a right triangle is equal to the ratio of the hypotenuse OB to adjacent leg OA.
Designations: sec, sec α, sec xFor example, if the adjacent leg is 2 and the hypotenuse is 4, then angle α will be equal to 60°, and secant of angle α will be 4/2=2
As you can see, if the angle tends to 90°, then the leg OA will tend to 0 and secant of the angle in this limiting case will tend to infinity and in the end it will not be determined: OB/OA = OB/0 = X (±∞).
And in the same way, if the angle tends to 0, then the length of the adjacent leg will tend to equal the length of the hypotenuse and the secant of the angle of such an angle in the limiting case will be equal to 1: OB/OA = OB/OB = 1.
Secant of an arbitrary angle
The secant of an angle in a right triangle is the secant of an acute angle, a special case of the secant of an arbitrary angle
Let us draw an additional vertical axis parallel to the ordinate axis through the point (1; 0) - the tangent axis. And through the center of the coordinates, parallel to the radius, is the secant axis. The secant axis is dynamic, rotating along with the radius (hypotenuse) when the angle changes. The positive direction of the secant axis is aligned with the radius. Secant of an arbitrary angle (acute, right, obtuse, straight, concave, full) - the length of a segment with the beginning at the coordinate center (at the center of the circle) - at the zero of the secant axis, and the end at the intersection point of the secant axis and tangent axes.
What is the relationship between the secant of an arbitrary angle and the secant 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 secant 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 (secant axis) and the tangent axis.
From this it follows that:
1. Angle secant values from -90° to 90° (or from 270° to 360° and from 0° to 90°) are positive in the segment [1, ∞]
1. Angle secant values from -90° to 90° (or from 270° to 360° and from 0° to 90°) are positive in the segment [1, ∞]
2. The values of the secant angles from 90° to 270° are negative in the segment [-∞, -1]
3. And, summing up, the entire range of values [-∞; -1]⋃[1; ∞]
4. Angle values from 0° to 180° mirror the angle values from 180° to 360°
Secant values of some angles.
where N is the number of the finger starting from the thumb. The exception is the thumb (90°), for which the value is not defined
Mnemonic rule:
`sec α = 2/sqrt(N), `
| 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 | ind | `2` | `sqrt(2)` | `(2sqrt(3))/3` | 1 |
Equation
The equation sec x = a means that you need to find angles x whose secant is equal to a. For example the equation
`sec x = 1`
has solutions `x=2pik, k∈Z`
That is, x is an infinite series of values `... -4pi, -2pi, 0, 2pi, 4pi ...`
And the equation
`sec x = 2`
has solutions `x=(-1)^(k+1)*pi/6+pi/2+pik, k∈Z`
That is, x is an infinite series of values `... -(5pi)/3, -pi/3, pi/3, (5pi)/3, (7pi)/3, (11pi)/3 ...`
Properties
Сompare properties
Function
`y = sec x`
The secant function is inversely proportional to the cosine function.
Function domain
`D(sec x) = R - {-pi/2 + pin, (n in Z)}`
The domain of definition of the secant is the entire numerical axis, with the exception of points at which the cosine 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 II quarters and between the III and IV quarters.
Range
`E(sec x) = [-infty, -1] uu [1, infty]`
The range of secant values is the entire numerical axis, with the exception of the interval from -1 to 1. The secant simply cannot become less than 1 in absolute value, since it is inversely proportional to the cosine, which cannot be greater than 1 in absolute value.
Parity
`even`
Like cosine, secant is an even function. When the angle value passes through 0, it does not change its sign. The function is symmetrical about the ordinate axis (y-axis).
Periodicity
`T = 2pi`
The trajectory of the secant function begins to repeat only after completing a full circle.
Zeros
`no`
Based on the range of values and the graph, it is clear that the secant function has no values between -1 and 1 and does not intersect the horizontal x-axis.
Constant sign intervals
`sec x < 0,`
`x in (pi/2+2pin, (3pi)/2+ 2pin), n in Z`
`sec x > 0,`
`x in (-pi/2 + 2pin, pi/2 + 2pin), n in Z`
The secant function is negative in the II and III quarters and positive in the I and IV quarters of the circle.
Monotony intervals
`uarruarruarr x in [2pin, pi/2 + 2pin)uu`
`uu(pi/2+2pin, pi+2pin], n in Z`
`darrdarrdarr x in [-pi+2pin, -pi/2+2pin)uu`
`uu(-pi/2+2pin, 2pin], n in Z`
The secant function decreases in the III and IV quarters, and increases in the I and II quarters of the circle.
Extrema
`y_min = 1, x = 2pin, n in Z`
`y_max = -1, x = pi+2pin, n in Z`
The secant function has local minima equal to 1 between the I and IV quarters and local maxima equal to -1 between the II and III quarters of the circle.
Inflection points
`no`
It is nowhere possible to draw a tangent to the secant function such that its local neighborhoods are on opposite sides of this tangent.
Formulas
`sec alpha = 1/cos alpha`
`sec alpha = (tan alpha)/sin alpha`
`sec alpha = text(csc)(alpha+pi/2) = - text(csc)(alpha-pi/2)`
`sec alpha = sec(-alpha)`
`(sec alpha)^' =sin alpha/cos^2 alpha`
`int sec (alpha)dalpha = -ln |tan(pi/2 + alpha/2)| + C`