Cosine
In a right triangle, the cosine of an angle θ is defined as the ratio of the adjacent leg (b) to the hypotenuse (c).
It is one of the trigonometric ratios. They are called ratios because they are expressed as the quotient of two of the sides of a right triangle.
Its abbreviation is cos (from the Latin cosinus).
Cosines for Special Common Angles
The following table gives the values of cosines for common angles:
Properties of Cosine
- Domain: (all real numbers).
- Range:
- Symmetry: since cos (-x) = cos (x) then cos (x) is an even function and its graph is symmetric with respect to the Y axis.
- Increasing-decreasing behaviour: over one period and from 0 to 2π, cos (x) is decreasing on (0, π) and increasing on the interval (π, 2π).
- End behaviour: The limits as x approaches ±∞ do not exist since the function values oscillate between +1 and −1. This is a periodic function with period 2π.
- The derivative of cosine function:
- The integral of cosine function:
Graphical Representation of the Cosine Function
The cosine is a periodic function with period 2π radians (360 degrees), so this section of the graph will be repeated in different periods.
Geometric Representation of the Cosine
Relationship Between Cosine and Other Trigonometric Functions
There are some basic trigonometric identities involving cosine:
- Relationship between cosine and sine:
- Relationship between cosine and tangent:
- Relationship between cosine and cosecant:
- Relationship between cosine and secant:
- Relationship between cosine and cotangent:
(1) Note: the sign depends on the quadrant of the original angle.
Trigonometric Identities Involving the Cosine Function
Cosine of Complementary, Supplementary and Conjugate Angles
- Cosine of a Complementary Angle:
- Cosine of a Supplementary Angle:
- Cosine of a Conjugate Angle:
Cosine of Negative Angles
- Cosine of a Negative Angle:
Cosine of Angles that Differs by 90º or 180º
- Cosine of an Angle that Differs by 90º:
- Cosine of an Angle that Differs by 180º:
Sum, Difference, Double-Angle, Half-Angle and Triple-Angle Identities for Cosine
- Angle Sum Identity for Cosine:
- Angle Difference Identity for Cosine:
- Double-Angle Identity for Cosine:
- Half-Angle Identity for Cosine:
- Triple-Angle Identity for Cosine:
Sum to Product and Product to Sum Identities for Cosine
These identities assist in transforming a trigonometric expression presented as a product to a sum or vice versa:
- Sum to Product Identities for Cosine:
- Product to Sum Identities for Cosine:
Law of Cosines
The law of cosines (or Cosine Rule) relates one side of a triangle to the other two and the angle they form. The theorem states that:
The square of any side (a, b or c) of a triangle is equal to the sum of the squares of the two remaining sides minus twice their product by the cosine of the angle (A, B or C) they form:
The law of cosines is a generalization of the Pythagorean theorem, which holds only for right triangles. In fact, if A is right angle (90º), its cosine is zero and hence, a^{2} = b^{2}+c^{2}.
If angle A is obtuse, that is >90º, then the cosine is negative.
Other Trigonometric Ratios
The trigonometric ratios of an angle θ are the ratios obtained between the three sides of a right triangle. That is the comparison by the quotient of its three sides a, b and c.
- The sine is the ratio between the length of the oppisite leg (a) divided by the length of the hypotenuse (c). In a formula, it is written simply as sin:
- The tangent is the ratio between the length of the opposite leg (a) divided by the length of the adjacent leg (b). In a formula, it is written simply as tan:
Trigonometric Ratios for Special Common Angles
The trigonometric ratios for the most common angles (0º, 30º, 45º, 60º, 90º, 180º and 270º) are:
Relationships Between Trigonometric Functions
Any trigonometric ratio can be expressed in terms of any other. The following table shows the formulas with which each one is expressed as a function of the other:
Note: the sign + or – depends on the quadrant of the original angle.
Reciprocal Trigonometric Ratios
Reciprocal trigonometric ratios are the multiplicative inverses of trigonometric ratios. These are:
- Cosecant (csc): It is the reciprocal ratio of the sine. That is, csc θ · sen θ = 1 or csc θ = 1/sen θ.
- Secant (sec): It is the reciprocal ratio of the cosine. That is, sec θ · cos θ = 1 or sec θ = 1/cos θ.
- Cotangent (cot): It is the reciprocal ratio of the tangent. Also in this case, cot θ · tan θ = 1 or cot θ = 1/tan θ.
Trigonometric Functions
Trigonometric functions are also called circular functions. The reason is that in a right triangle that has been drawn on the coordinate axis, with the vertex of the angle θ at the center of a circle (O), that is, in standard position, its vertex B can go through all the points of this circle.
Trigonometric functions and reciprocal trigonometric functions can be graphically represented in a right triangle on a circle of r=1.