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Example:Solving Trigonometric Equation

 Post to:   Explanation:

Trigonometric functions

In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications.

## Right-angled triangle definitions The notion that there should be some standard correspondence between the lengths of the sides of a triangle and the angles of the triangle comes as soon as one recognizes that similar triangles maintain the same ratios between their sides. That is, for any similar triangle the ratio of the hypotenuse (for example) and another of the sides remains the same. If the hypotenuse is twice as long, so are the sides. It is these ratios that the trigonometric functions express.

To define the trigonometric functions for the angle A, start with any right triangle that contains the angle A. The three sides of the triangle are named as follows:

§  The hypotenuse is the side opposite the right angle, in this case side h. The hypotenuse is always the longest side of a right-angled triangle.

§  The opposite side is the side opposite to the angle we are interested in (angle A), in this case side a.

§  The adjacent side is the side having both the angles of interest (angle A and right-angle C), in this case side b.

In ordinary Euclidean geometry, according to the triangle postulate the inside angles of every triangle total 180°  radians). Therefore, in a right-angled triangle, the two non-right angles total 90° (π/2 radians), so each of these angles must be in the range of (0˚,90°) as expressed in interval notation. The following definitions apply to angles in this 0° - 90° range. They can be extended to the full set of real arguments by using the unit circle, or by requiring certain symmetries and that they be periodic functions. For example, the figure shows sin θ for angles θ, π − θ, π + θ, and 2π − θ depicted on the unit circle (top) and as a graph (bottom). The value of the sine repeats itself apart from sign in all four quadrants, and if the range of θ is extended to additional rotations, this behavior repeats periodically with a period 2π.

The trigonometric functions are summarized in the following table and described in more detail below. The angle θ is the angle between the hypotenuse and the adjacent line - the angle at A in the accompanying diagram.

 Function Abbreviation Description Identities  (using radians) Sine sin opposite / hypotenuse Cosine cos adjacent / hypotenuse Tangent tan (or tg) opposite / adjacent Cotangent cot (or ctg or ctn) adjacent / opposite Secant sec hypotenuse / adjacent Cosecant csc (or cosec) hypotenuse / opposite http://en.wikipedia.org/wiki/Trigonometric_functions

# Inverse trigonometric functions

In mathematics, the inverse trigonometric functions or cyclometric functions are the inverse functions of the trigonometric functions, though they do not meet the official definition for inverse functions as their ranges are subsets of the domains of the original functions. Since none of the six trigonometric functions are one-to-one (by failing the horizontal line test), they must be restricted in order to have inverse functions.

For example, just as the square root function is defined such that y2 = x, the function y = arcsin(x) is defined so that sin(y) = x. There are multiple numbers y such that sin(y) = x; for example, sin (0) = 0, but also sin (π) = 0, sin (2π) = 0, etc. It follows that the arcsine function is multivalued: arcsin (0) = 0, but also arcsin (0) = π, arcsin (0) = 2π, etc. When only one value is desired, the function may be restricted to its principal branch. With this restriction, for each x in the domain the expression arcsin(x) will evaluate only to a single value, called its principal value. These properties apply to all the inverse trigonometric functions.

The principal inverses are listed in the following table.

 Name Usual notation Definition Domain of x for real result Range of usual principal value (radians) Range of usual principal value (degrees) arcsine y = arcsin x x  = sin y −1 ≤ x ≤ 1 −π/2 ≤ y ≤ π/2 −90° ≤ y ≤ 90° arccosine y = arccos x x = cos y −1 ≤ x ≤ 1 0 ≤ y ≤ π 0° ≤ y ≤ 180° arctangent y = arctan x x= tan y all real numbers −π/2 < y < π/2 −90° < y < 90° arccotangent y = arccot x x = cot y all real numbers 0 < y < π 0° < y < 180° arcsecant y = arcsec x x = sec y x ≤ −1 or 1 ≤ x 0 ≤ y < π/2 or π/2 < y ≤ π 0° ≤ y < 90° or 90° < y ≤ 180° arccosecant y = arccsc x x = csc y x ≤ −1 or 1 ≤ x −π/2 ≤ y < 0 or 0 < y ≤ π/2 -90° ≤ y < 0° or 0° < y ≤ 90°

## General solutions

Each of the trigonometric functions is periodic in the real part of its argument, running through all its values twice in each interval of 2π. Sine and cosecant begin their period at 2πk − π/2 (where k is an integer), finish it at 2πk + π/2, and then reverse themselves over 2πk + π/2 to 2πk + 3π/2. Cosine and secant begin their period at 2πk, finish it at 2πk + π, and then reverse themselves over 2πk + π to 2πk + 2π. Tangent begins its period at 2πk − π/2, finishes it at 2πk + π/2, and then repeats it (forward) over 2πk + π/2 to 2πk + 3π/2. Cotangent begins its period at 2πk, finishes it at 2πk + π, and then repeats it (forward) over 2πk + π to 2πk + 2π.

This periodicity is reflected in the general inverses where k is some integer:      (Our solved example in mathguru.com uses this concept)

http://en.wikipedia.org/wiki/Inverse_trigonometric_functions#General_solutions