# Trigonometric Expressions

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In Trigonometry an expression means a statement with all possible combinations of variables, constants and operators. A trigonometric expression means an expression which contains at least one trigonometric ratio.

Like general expressions trigonometric expressions can also undrgo all operations. They can be simplified, factored and evaluated. A trigonometric expression may also mean an identity. For example sin2x + cos2x is always 1 for all real values of x.

In this section we will study some of the trigonometric expressions.

## Trigonometric Expressions – with Single Trigonometric Ratio

A trigonometric expression may sometimes have only one trigonometric ratio. Because of the inter relations between all the trigonometric ratios, it is possible to change a given expression with any trigonometric ratio into expressions all other trigonometric ratios.

The following illustration is an example of expressing in terms of all trigonometric ratios in terms of the sine rule ratio of an angle.

## Trigonometric Expressions – in Terms of Different Ratios

Look at the following expressions.

(Sin A)(Cos B) + (Cos A)(Sin B)

(Cos A Cos B) – (Sin A Sin B)

$\frac{(Tan A + Tan B)}{1 – (Tan A)(Tan B)}$

Each one of the above convey another expression.

The above expressions are same as Sin(A + B), Cos(A + B) and Tan (A + B) respectively.

## Trigonometric Expressions – Used in Identities

We are very much aware that sin2x + cos2x is always 1 for all real values of x.

There are many more examples that some trigonometric expressions mean identities or the expression is a constant when evaluated for any real value of the variable.

The following are the examples related to any triangle.

( Assume the letters A, B, and C represent the measures of the internal angles of the triangle and a, b and c are the measures of the corresponding sides)

a = (c)(Cos B) + (b)(Cos C) (the same cycle follows for other sides)

a2 = b2 + c2 – (2bc)(Cos A) (the same cycle follows for other sides)

$\frac{a}{Sin A}$ = $\frac{b}{Sin B}$ = $\frac{c}{Sin C}$

In three dimensional geometry, if a line makes an angle ?, ? and ?, with x-axis, y-axis and z-axis respectively, the value of the expression

cos2? + cos2? + cos2?

is always 1.

## Related Tags

Concepts of Trigonometric expressions, What are Trigonometric expressions?, Concepts on Trigonometric expressions