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Chapter 9

Getting Started with Trig Identities

Don’t have an identity crisis! In this chapter, you become more familiar with the possibilities for rewriting trigonometric expressions. A trig identity is really an equivalent expression or form of a function that you can use in place of the original. The equivalent format may make factoring easier, solving an application possible, and (later) performing an operation in calculus more manageable.

The trigonometric identities are divided into many different classifications. These groupings help you remember the identities and make determining which identity to use in a particular substitution easier. In a classic trig identity problem, you try to make one side of the equation match the other side. The best way to do so is to work on just one side — the left or the right — but sometimes you need to work on both sides to see just how to work the problem to the end.

The Problems You’ll Work On

In this chapter, you’ll work with the basic trigonometric identities in the following ways:

 Determining which trig functions are reciprocals of one another

 Creating Pythagorean identities from a right triangle whose hypotenuse measures 1 unit

 Determining the sign of identities whose angle measure is negated

 Matching up trig functions and their co-functions

 Using the periods of functions in identities

 Making the most of selected substitutions into identities

 Working on only one side of the identity

 Figuring out where to go with an identity by working both sides at once

What to Watch Out For

Don’t let common mistakes trip you up; keep in mind that when working on trigonometric identities, some challenges will include the following:

 Keeping track of where the 1 goes in the Pythagorean identities

 Remembering the middle term when squaring binomials involving trig functions

 Correctly rewriting Pythagorean identities when solving for a squared term

 Recognizing the exponent notation

Proving Basic Trig Identities

511–535 Prove the trig identity. Indicate your first identity substitution.

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512.

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Returning to Basic Sine and Cosine to Solve Identities

536–540 Determine the missing term or factor in the identity by changing all functions to those using sine or cosine.

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540.

Using Multiplication by a Conjugate to Solve Identities

541–545 Determine the missing term or factor in the identity by multiplying by a conjugate.

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Solving Identities After Raising a Binomial to a Power

546–550 Determine the missing term or terms in the identity after computing the power of the binomial.

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Solving Identities After Factoring out a Common Function

551–555 Determine the missing term or factor in the identity after factoring.

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Solving Identities After Combining Fractions

556–560 Determine the missing term or factor in the identity after adding the fractions.

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Performing Algebraic Processes to Make Identities More Solvable

561–570 Give the rewritten identity after performing the action. Then complete the solution of the identity.

561. Distribute on the right and move all terms to the left.

562. Split up the first fraction on the left and distribute on the right.

563. Cross-multiply.

564. Cross-multiply.

565. Factor the numerator on the left and reduce the fraction.

566. Cross-multiply.

567. Add the middle fraction to both sides, and then add the two fractions on the right. Simplify by using reciprocal identities.

568. Combine the two terms on the right.

569. Subtract the two terms on the left.

570. Square both sides of the equation.