Cosine
Cosine of an angle in a right triangle
In a right triangle OAB OA and AB are legs, OB is the hypotenuse.The cosine of an angle in a right triangle is equal to the ratio of the
Designations: cos, cos α, cos x
The cosine of an angle in a right triangle is equal to the ratio of the adjacent leg OA к гипотенузе OB.
Designations: cos, cos α, cos xFor example, if the adjacent leg is 2 and the hypotenuse is 4, then angle α will be equal to 60°, and cosine of angle α will be 2/4=0.5
As you can see, if the angle tends to 90°, then the leg OA will tend to 0 and cosine of the angle in this limiting case will be equal to 0: OA/OB = 0/OB = 0.
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 cosine of such an angle in the limiting case will be equal to 1: OA/OB = OB/OB = 1.
Cosine of an arbitrary angle
The cosine of an angle in a right triangle is the cosine of an acute angle, a special case of the cosine of an arbitrary angle
Cosine of an arbitrary angle (acute, right, obtuse, straight, concave, complete) is the abscissa of a point (horizontal coordinate x) obtained by rotating the point (1; 0) around the origin by an angle counterclockwise on the unit circle.
What is the relationship between the cosine of an arbitrary angle and the cosine in a right triangle? The radius OB, its projection OA on the x-axis and the segment AB connecting the radius and the projection form such a right triangle. The radius is the hypotenuse, its projection OB is the adjacent side, segment AB is the opposite side. As you can see, the segment OB is the projection of the radius onto the x axis. And the cosine of the angle will be the ratio of the length of the segment OB to the hypotenuse. If we consider a circle with radius 1, then the hypotenuse is equal to 1 and the cosine of the angle is equal to the length of the segment OB or the abscissa of a point on the circle, its coordinate along the x axis.
From this it follows that:
1. The cosine values of angles from -90° to 90° (or from 270° to 360° and from 0° to 90°) are positive in the segment [0, 1]
1. The cosine values of angles from -90° to 90° (or from 270° to 360° and from 0° to 90°) are positive in the segment [0, 1]
2. The cosine values of angles from 90° to 270° are negative in the segment [-1, 0]
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°
5. Cosine axis - horizontal axis, abscissa axis, x axis
Cosine values of some angles.
where N is the number of the finger starting from the thumb.
Mnemonic rule:
`cos α = sqrt(N)/2, `
| 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 | `1/2` | `sqrt(2)/2` | `sqrt(3)/2` | 1 |
Equation
Equation cos x = a means that you need to find angles x whose cosine is equal to a. For example the equation
`cos 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
`cos x = 0.5`
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 = cos x`
The cosine function is inversely proportional to the secant function.
Function domain
`D(sin x) = R`
The domain of definition of the cosine is the entire numerical axis.
Range
`E(sin x) = [-1, 1]`
The range of values of the cosine is the interval from -1 to 1. This happens because the cosine is the ratio of the leg to the hypotenuse, and the leg in a triangle is always less, or, in the extreme case, equal to the hypotenuse.
Parity
`even`
When the angle value passes through 0, the cosine function of this angle does not change its sign. The function is symmetrical about the ordinate axis (y-axis).
Periodicity
`T = 2pi`
The trajectory of the cosine function begins to repeat only after completing a full circle.
Zeros
`cos x = 0, x=pi/2+pin, n in Z`
The cosine function has zero values at points between the I and II quarters and between the III and IV quarters of the circle.
Constant sign intervals
`cos x > 0, x in (-pi/2 + 2pin, pi/2 + 2pin), n in Z`
`cos x < 0, x in (pi/2 + 2pin, (3pi)/2 + 2pin), n in Z`
The cosine 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 [-pi + 2pin, 2pin], n in Z`
`darrdarrdarr x in [2pin, pi + 2pin], n in Z`
The cosine function increases in the III and IV quarters, and decreases in the I and II quarters of the circle.
Extrema
`y_min = -1, x = pi+2pin, n in Z`
`y_max = 1, x = 2pin, n in Z`
The cosine function has local minima equal to -1 between the II and III quarters and local maxima equal to 1 between the I and IV quarters of the circle.
Inflection points
`y=0, x = pi/2 + pin, n in Z`
The cosine function has inflection points between the I and II quarters and between the III and IV quarters of the circle with a value of 0.
Formulas
`cos alpha = sin(alpha + pi/2)`
`cos alpha = +- sqrt(1 - sin^2alpha)`
`cos alpha = sin alpha * text(cot α)`
`cos alpha = sin alpha / text(tan α)`
`cos alpha = cos(-alpha)`
`(cos alpha)^' =- sin alpha`
`int cos (alpha)dalpha = sin alpha + C`