# The Mathematics

## Double- , triple-, and half-angle formulas

Double- , triple-, and half-angle formulas Signs before the roots are selected depending on the quarter, in which the angle is placed. Back The math online

## Transforming of trigonometric expressions to product

Transforming of trigonometric expressions to product Back The math online

## Some important correlations

Some important correlations Transforming product-to-sum. Universal substitution. Quadruple-angle formulas. The last three formulas are called a universal substitution; they are used at solving some of trigonometric equations and integrating of trigonometric functions. Back The math online

## Basic relations between elements of triangle

Law of cosines. Law of sines. Law of tangents. Area formulas. Heron’s  formula. Radii of circumscribed and inscribed circles. Designations: a, b, c – sides;  A, B, C – angles of triangle; p = ( a + b + c ) / 2  – a half-perimeter;  h – a height;  S – an area;  R, r  …

## Solving of oblique triangles – The Math

Case 1. Three sides  a, b, c  are given. Find angles A, B, C. By the law of cosines we find one of the angles:the second angle we find by the law of sines:the third angle is found by the formula:  C = 180° – ( A + B ). E x a m p …

## Inverse trigonometric functions – The Math

Inverse trigonometric functions. Multiple-valued functions. Principal values of inverse trigonometric functions. The relation  x = sin y  permits to find both  x  by the given  y , and also  y by the given  x ( at  | x | 1 ). So, it is possible to consider not only a sine as a functionof an angle, …

## Basic relations for inverse trigonometric functions

Basic relations for inverse trigonometric functions Back The math online

## Trigonometric equations. Main methods for solving

Trigonometric equations. Simplest trigonometric equations. Methods of solving:  algebraic method,  factoring, reducing to a homogeneous equation, transition to a half-angle, introducing an auxiliary angle, transforming a product to a sum, universal substitution. Trigonometric equations. An equation, containing an unknown under the trigonometric function sign is called trigonometric. Simplest trigonometric equations. Methods for solving trigonometric equations. …

## Systems of simultaneous trigonometric equations

At solving simultaneous trigonometric equations we use the same methods, as in algebra ( exchange, substitution, elimination etc.) and also the knownmethods and formulas of trigonometry. Consider some examples. E x a m p l e  1 .   Solve the system of simultaneous equations: E x a m p l e   2 .  Solve the …

## Trigonometric inequalities – The Math

At solving of trigonometric inequalities we use the properties of inequalities, known from algebra and also the trigonometric transformations and formulas.Using of the unit circle at solving of trigonometric inequalities is almost necessary. Consider some examples. E x a m p l e  1 .  Solve the inequality:   sin x > 0. S o l …