Trigonometric Identities and Function Properties

Even and Odd Trigonometric Functions

Understanding the symmetry properties of trigonometric functions is essential for calculus applications. A function is even if and odd if .

• Sine: (odd function)

• Cosine: (even function)

• Tangent: (odd function)

• Secant: (even function)

• Cosecant: (odd function)

• Cotangent: (odd function)

Example: Show that the tangent function is odd:

Example: Show that the secant function is even:

Basic Trigonometric Identities

Addition and Subtraction Formulas

These formulas allow you to express trigonometric functions of sums or differences of angles in terms of functions of individual angles.

Double-Angle Formulas

Useful Rewriting for Integrals

Complementary Angle Identities

Note: These can be visualized using right triangles and the SOH-CAH-TOA definitions.

Derivatives of Trigonometric Functions

Special Limits for Trigonometric Derivatives

Two fundamental limits are used in deriving the derivatives of sine and cosine:

Derivative of Sine

• Using the definition of the derivative and the sine addition formula:

$$ \frac{d}{dx}\sin x=\cos x $$

Derivative of Cosine

• Similarly, using the definition of the derivative:

$

Derivative of Tangent

• Using the quotient rule:

$

Derivative of Cotangent

• Using the quotient rule:

$

Derivative of Secant

• Using the chain and quotient rules:

$

Derivative of Cosecant

• Using the chain and quotient rules:

$

Summary Table: Derivatives of Trigonometric Functions

Example: Derivative of a Composite Function

Find the derivative of :

• Apply the chain rule:

$

L'Hospital's Rule and Indeterminate Forms

Indeterminate Forms

When evaluating limits, you may encounter expressions like or , which are called indeterminate forms. L'Hospital's Rule provides a method for evaluating such limits.

L'Hospital's Rule

• If and , or both approach , then:

$

provided the limit on the right exists.

Examples Using L'Hospital's Rule

• : Both numerator and denominator approach 0, so apply L'Hospital's Rule:

$

• : Both approach 0, so apply L'Hospital's Rule:

$

• : Both approach , so apply L'Hospital's Rule repeatedly:

$

Notes on L'Hospital's Rule

• Always check that the conditions for L'Hospital's Rule are satisfied before applying it.

• Take derivatives of the numerator and denominator separately; do not use the quotient rule.

• Sometimes, algebraic simplification is easier than applying L'Hospital's Rule.

Summary Table: Indeterminate Forms and L'Hospital's Rule

Key Takeaways

• Know the basic trigonometric identities and properties of even/odd functions.

• Be able to compute derivatives of all six trigonometric functions.

• Understand and apply L'Hospital's Rule to evaluate limits involving indeterminate forms.

• Always verify the conditions for L'Hospital's Rule before using it.