Section 6.1 Basic Identities

Definition of an Identity

An identity is an equation that is always true for all values of the variable for which each expression is defined. Identities are fundamental in mathematics as they allow us to simplify and manipulate expressions.

• Example a: (true for all real numbers)

• Example b: (true for all real numbers)

• Example c: (true for all real numbers except )

Trigonometric identities are equations involving trigonometric functions that are true for all values in their domains.

Basic Trigonometric Identities

• Quotient Identities:

• Reciprocal Identities:

• Pythagorean Identities:

Even and Odd Trigonometric Functions

Trigonometric functions can be classified as even or odd based on their symmetry properties:

• Even Functions: (symmetric about the y-axis)

  • Cosine:

  • Secant:

• Odd Functions: (symmetric about the origin)

  • Sine:

  • Cosecant:

  • Tangent:

  • Cotangent:

Examples:

• (odd)

• (even)

Section 6.2 Verifying Identities

Algebraic Techniques for Verifying Identities

To verify an identity, manipulate one side of the equation using algebraic and trigonometric properties until it matches the other side. Key strategies include:

• Work with the more complicated side first.

• Rewrite all trigonometric functions in terms of sine and cosine when possible.

• Use even/odd properties to simplify negative angles.

• Apply Pythagorean and other basic identities.

• Factor expressions, especially using the difference of squares.

• Split or combine rational expressions as needed.

Example: Verify

• Rewrite the right side as

• Combine and manipulate using identities and algebra to show both sides are equivalent.

Algebra Review for Verifying Identities

• Multiplying Polynomials:

• Factoring Polynomials:

• Simplifying Expressions:

Section 6.3 Sum and Difference Identities

Angle Sum and Difference Identities

These identities allow us to express the sine, cosine, or tangent of a sum or difference of two angles in terms of the sines and cosines of the individual angles.

• Cosine of a Sum:

• Sine of a Sum:

• Tangent of a Sum:

• Cosine of a Difference:

• Sine of a Difference:

• Tangent of a Difference:

Example:

Cofunction Identities

Cofunction identities relate the trigonometric functions of complementary angles (angles that add up to or radians):

These identities also hold for degree measures, replacing with .

Section 6.4 Double Angle and Half Angle Identities

Double Angle Identities

Double angle identities express trigonometric functions of in terms of :

• Alternative forms for cosine:

Half Angle Identities

Half angle identities allow us to find the sine, cosine, or tangent of half an angle:

Example: To find , use the half-angle identity:

Section 6.6 Conditional Trigonometric Equations

Conditional Equations

A conditional equation is true only for specific values of the variable. For example, is true only for certain values of .

• Solutions can be found graphically (by intersection points) or algebraically.

• There are generally infinitely many solutions due to the periodic nature of trigonometric functions.

• General solution for : and , for any integer .

• On the interval , the solutions are and .

Strategies for Solving Conditional Trigonometric Equations

• Isolate the trigonometric function.

• Use algebraic techniques (factoring, quadratic formula, etc.) as appropriate.

• Apply inverse trigonometric functions to find angle solutions.

• Consider all possible solutions within the given interval and the periodicity of the function.

• Check for extraneous solutions, especially in rational and radical equations.

Types of Trigonometric Equations and Solution Strategies

Sample Solutions (in radians, [0, 2π) and all solutions)

Additional Strategies

• When multiple trigonometric functions appear, try to rewrite all terms in terms of a single function using identities.

• Use double angle, sum/difference, or Pythagorean identities as needed to simplify.

• For equations with coefficients on the variable, remember to adjust the period when writing general solutions.

• Always check for extraneous solutions, especially when squaring both sides or dealing with denominators.