Verifying Verifying Trigonometric Trigonometric

Identities Identities Section 5.2 Section 5.2

Math 1113 Math 1113

Created & Presented by Laura Created & Presented by Laura Ralston Ralston

Verifying Trigonometric Verifying Trigonometric Identities Identities

In this section, we will study In this section, we will study techniques for verifying techniques for verifying trigonometric identities.trigonometric identities.

The key to verifying identities is the The key to verifying identities is the ability to use the fundamental ability to use the fundamental identities and rules of algebra to identities and rules of algebra to rewrite trigonometric expressionsrewrite trigonometric expressions

Review Review

Algebraic Algebraic ExpressionExpression: a : a collection of collection of numbers, variables, numbers, variables, symbols for symbols for operations, and operations, and grouping symbols; grouping symbols; contains no equal contains no equal signsign

2(4x -3) – 6 2(4x -3) – 6

2sin(4x – 2sin(4x – ) + 3 ) + 3

EquationEquation: a : a statement that two statement that two mathematical mathematical expressions are expressions are equal. equal.

x + 2 = 5 x + 2 = 5

sin x = 0 sin x = 0

Three Categories of Three Categories of EquationsEquations

ContradictionContradiction: no : no values of the values of the variable make the variable make the equation true equation true

x + 2 = x x + 2 = x

sin x = 5 sin x = 5

ConditionalConditional: only 1 : only 1 or several values of or several values of the variable make the variable make the equation truethe equation true

x + 2 = 5 ---- x = 3 x + 2 = 5 ---- x = 3

sin x = 0 ----- x = 0sin x = 0 ----- x = 0

IdentityIdentity: equation is true for EVERY : equation is true for EVERY value of the variable value of the variable

x + x = 2x x + x = 2x coscos22x + sinx + sin22x = x = 11

2x = 2x 2x = 2x

Verifying an Identity is quite different Verifying an Identity is quite different from solving an equation. from solving an equation.

There is no well-defined set of rules There is no well-defined set of rules to follow in verifying trigonometric to follow in verifying trigonometric identities and the process is best identities and the process is best learned by practice!!!learned by practice!!!

Guidelines for Verifying Trig Guidelines for Verifying Trig Identities Identities

Work with one side of the identity at Work with one side of the identity at a time. It is often better to work with a time. It is often better to work with the more complicated side first. the more complicated side first.

Look for opportunities to factor an Look for opportunities to factor an expression, add fractions, square a expression, add fractions, square a binomial, or create a monomial binomial, or create a monomial denominator. denominator.

Look for opportunities to use the Look for opportunities to use the fundamental identities. Note which fundamental identities. Note which functions are in the final expression functions are in the final expression you want. you want.

Sines and cosines pair up well, as do Sines and cosines pair up well, as do secants and tangents, and cosecants secants and tangents, and cosecants and cotangents. and cotangents.

If all else fails, try converting all If all else fails, try converting all terms to sines and cosines. terms to sines and cosines.

Always try something!! Even paths Always try something!! Even paths that lead to dead ends give you that lead to dead ends give you insights. insights.

There can be more than one way to There can be more than one way to verify an identity. Your method may verify an identity. Your method may differ from that used by your differ from that used by your instructor or classmates. instructor or classmates.

This is a good chance to be creative This is a good chance to be creative

and establish your own style, but try and establish your own style, but try to be as efficient as possible. to be as efficient as possible.