Formulas and Tables

Trigonometry

Definition and Graphs of Trigonometric Functions

Angles (arguments of functions): \(\alpha\), \(x\)
Trigonometric functions: \(\sin \alpha \), \(\cos \alpha \), \(\tan \alpha \), \(\cot \alpha \), \(\sec \alpha \), \(\csc \alpha \)
Set of real numbers: \(\mathbb{R}\)
Coordinates of points on a circle: \(x\), \(y\)

Radius of a circle: \(r\)
Integers: \(k\)

  1. Trigonometric functions are elementary functions, the argument of which is an angle. Trigonometric functions describe the relation between the sides and angles of a right triangle. Applications of trigonometric functions are extremely diverse. For example, any periodic processes can be represented as a sum of trigonometric functions (Fourier series). These functions often appear in the solution of differential equations and functional equations.
  2. The trigonometric functions include the following \(6\) functions: sine, cosine, tangent, cotangent, secant, and cosecant. For each of these functions, there is an inverse trigonometric function.
  3. The trigonometric functions can be defined using the unit circle. The figure below shows a circle of radius \(r = 1\). There is a point \(M\left( {x,y} \right)\) on the circle. The angle between the radius vector \(OM\) and the positive direction of the \(x\)-axis is equal to \(\alpha\).
The sine and cosine of an angle alpha
  1. The sine of an angle \(\alpha\) is the ratio of the \(y\)-coordinate of the point \(M\left( {x,y} \right)\) to the radius \(r\):
    \(\sin \alpha = y/r\).
    Since \(r = 1\), the sine is equal to the \(y\)-coordinate of the point \(M\left( {x,y} \right)\).
  2. The cosine of an angle \(\alpha\) is the ratio of the \(x\)-coordinate of the point \(M\left( {x,y} \right)\) to the radius \(r\):
    \(\cos \alpha = x/r\)
  3. The tangent of an angle \(\alpha\) is the ratio of the \(y\)-coordinate of the point \(M\left( {x,y} \right)\) to the \(x\)-coordinate:
    \(\tan \alpha = y/x,\;\) \(x \ne 0\)
  4. The cotangent of an angle \(\alpha\) is the ratio of the \(x\)-coordinate of the point \(M\left( {x,y} \right)\) to the \(y\)-coordinate:
    \(\cot \alpha = x/y,\;\) \(y \ne 0\)
  5. The secant of an angle \(\alpha\) is the ratio of the radius \(r\) to the \(x\)-coordinate of the point \(M\left( {x,y} \right)\):
    \(\sec \alpha = r/x = 1/x,\;\) \(x \ne 0\)
  6. The cosecant of an angle \(\alpha\) is the ratio of the radius \(r\) to the \(y\)-coordinate of the point \(M\left( {x,y} \right)\):
    \(\csc \alpha = r/y = 1/y,\;\) \(y \ne 0\)
  7. Relation between the sides and angles in a right triangle
    In a unit circle, the projections \(x\), \(y\) of a point \(M\left( {x,y} \right)\) and the radius \(r\) form a right-angled triangle, in which \(x,y\) are the legs, and \(r\) is the hypotenuse. Therefore, the definitions given above are stated as follows:
    The sine of an angle \(\alpha\) is the ratio of the opposite leg to the hypotenuse.
    The cosine of an angle \(\alpha\) is the ratio of the adjacent leg to the hypotenuse.
    The tangent of an angle \(\alpha\) is the ratio of the opposite leg to the adjacent leg.
    The cotangent of an angle \(\alpha\) is the ratio of the adjacent leg to the opposite leg.
    The secant of an angle \(\alpha\) is the ratio of the hypotenuse to the adjacent leg.
    The cosecant of an angle \(\alpha\) is the ratio of the hypotenuse to the opposite leg.
  8. Graph of the sine function
    \(y = \sin x\), domain: \(x \in \mathbb{R}\), range: \(-1 \le \sin x \le 1\)
Graph of the sine function
  1. Graph of the cosine function
    \(y = \cos x\), domain: \(x \in \mathbb{R}\), range: \(-1 \le \cos x \le 1\)
Graph of the cosine function
  1. Graph of the tangent function
    \(y = \tan x\), domain: \(x \in \mathbb{R},\) \( x \ne \left( {2k + 1} \right)\pi/2\), range: \(- \infty \lt \tan x \lt \infty\)
Graph of the tangent function
  1. Graph of the cotangent function
    \(y = \cot x\), domain: \(x \in \mathbb{R},\) \(x \ne k\pi\), range: \(- \infty \lt \cot x \lt \infty\)
Graph of the cotangent function
  1. Graph of the secant function
    \(y = \sec x\), domain: \(x \in \mathbb{R},\) \(x \ne \left( {2k + 1} \right)\pi/2\), range: \(\sec x \in \) \(\left( {-\infty , -1} \right] \cup \left[ {1,\infty } \right)\)
Graph of the secant function
  1. Graph of the cosecant function
    \(y = \csc x\), domain: \(x \in \mathbb{R},\) \( x \ne k\pi\), range: \(\csc x \in \) \(\left( {-\infty , -1} \right] \cup \left[ {1,\infty } \right)\)
Graph of the cosecant function