Verifying Trigonometric Identities Section 5.2 Math 1113 Created & Presented by Laura Ralston

Verifying Trigonometric Identities In this section, we will study techniques for verifying trigonometric identities. In this section, we will study techniques for verifying trigonometric identities. The key to verifying identities is the ability to use the fundamental identities and rules of algebra to rewrite trigonometric expressions The key to verifying identities is the ability to use the fundamental identities and rules of algebra to rewrite trigonometric expressions

Review Algebraic Expression: a collection of numbers, variables, symbols for operations, and grouping symbols; contains no equal sign Algebraic Expression: a collection of numbers, variables, symbols for operations, and grouping symbols; contains no equal sign 2(4x -3) – 6 2sin(4x –  ) + 3 Equation: a statement that two mathematical expressions are equal. Equation: a statement that two mathematical expressions are equal. x + 2 = 5 x + 2 = 5 sin x = 0 sin x = 0

Three Categories of Equations Contradiction: no values of the variable make the equation true Contradiction: no values of the variable make the equation true x + 2 = x x + 2 = x sin x = 5 sin x = 5 Conditional: only 1 or several values of the variable make the equation true Conditional: only 1 or several values of the variable make the equation true x + 2 = x = 3 x + 2 = x = 3 sin x = x = 0 sin x = x = 0

Identity: equation is true for EVERY value of the variable Identity: equation is true for EVERY value of the variable x + x = 2x cos 2 x + sin 2 x = 1 x + x = 2x cos 2 x + sin 2 x = 1 2x = 2x 2x = 2x

Verifying an Identity is quite different from solving an equation. Verifying an Identity is quite different from solving an equation. There is no well-defined set of rules to follow in verifying trigonometric identities and the process is best learned by practice!!! There is no well-defined set of rules to follow in verifying trigonometric identities and the process is best learned by practice!!!

Guidelines for Verifying Trig Identities Work with one side of the identity at a time. It is often better to work with the more complicated side first. Work with one side of the identity at a time. It is often better to work with the more complicated side first. Look for opportunities to factor an expression, add fractions, square a binomial, or create a monomial denominator. Look for opportunities to factor an expression, add fractions, square a binomial, or create a monomial denominator.

Look for opportunities to use the fundamental identities. Note which functions are in the final expression you want. Look for opportunities to use the fundamental identities. Note which functions are in the final expression you want. Sines and cosines pair up well, as do secants and tangents, and cosecants and cotangents. Sines and cosines pair up well, as do secants and tangents, and cosecants and cotangents.

If all else fails, try converting all terms to sines and cosines. If all else fails, try converting all terms to sines and cosines. Always try something!! Even paths that lead to dead ends give you insights. Always try something!! Even paths that lead to dead ends give you insights.

There can be more than one way to verify an identity. Your method may differ from that used by your instructor or classmates. There can be more than one way to verify an identity. Your method may differ from that used by your instructor or classmates. This is a good chance to be creative and establish your own style, but try to be as efficient as possible. This is a good chance to be creative and establish your own style, but try to be as efficient as possible.