Calculus I: Trigonometric Functions &
Applications
Study Notes
Introduction
These notes systematically organize trigonometric knowledge for
Calculus I (Math 150A), bridging algebraic manipulation with
geometric intuition and real-world applications. The focus is on
mastering trigonometric identities and properties for calculus
problem-solving, emphasizing critical insights beyond basic formula
memorization.
Core Trigonometric Identities
Core Idea: Mastering trigonometric identities is essential for
simplification, differentiation, and integration of trigonometric
functions in calculus.
Pythagorean Identities: Fundamental identities relating squares
of sine, cosine, and tangent.
sin²θ + cos²θ = 1, tan θ = sinθ/cosθ.
Key Derivations: Useful for simplifying integrals and derivatives.
sin²θ = 1 - cos²θ, cos²θ = 1 - sin²θ.
Double-Angle Formulas: Express trigonometric functions of 2θ in
terms of θ.
sin2θ = 2sinθcosθ, cos2θ = cos²θ – sin²θ = 2cos²θ − 1, tan2θ =
2tanθ / (1 - tan²θ).
Power-Reduction Formulas: Useful for integrating sin²x or cos²x.
sin²θ = (1 - cos2θ)/2, cos²θ = (1 + cos2θ)/2.
Sum and Difference Formulas: Essential for evaluating
trigonometric functions of non-special angles.
cos(α + β) = cosαcosβ – sinαsinβ, cos(α – β) = cosαcosβ + sinαsinβ.
sin(α + β) = sinαcosβ + cosαsinβ, sin(α – β) = sinαcosβ – cosαsinβ.
Example 1.1: Simplifying Using Pythagorean Identity
, Problem: Simplify the expression
1− cos2 x
sinx
Solution:
By the Pythagorean identity, 1 −cos 2 x=sin 2 x .
Thus,
sin 2 x
=sinx
sinx
Key Terminology:
Pythagorean identity: A fundamental trigonometric identity
derived from the unit circle.
Simplification: Reducing expressions to their most compact
and interpretable form.
Properties of Sine and Cosine Functions
Core Idea: Understanding the properties of sine and cosine
functions is crucial for calculus analysis, including analyzing function
behavior and solving related problems.
Core Properties Summary: Key properties aligned with calculus
applications.
Domain: R (all real numbers) for both sinx and cosx.
Range: [-1, 1] for both sinx and cosx.
Periodicity: Minimum positive period of 2π for both sinx and cosx.
Parity: sin(–x) = –sinx (odd function), cos(–x) = cosx (even function).
Monotonic Intervals: Intervals of increasing and decreasing behavior.
Extrema: Maximum and minimum values and their locations.
Calculus-Focused Interpretations: Periodicity and Integration
Integrals over intervals of length 2π can be simplified due to
periodicity.
Monotonicity and Derivatives: The monotonic intervals relate
directly to the sign of the derivative.
For y = sinx, y' = cosx.
For y = cosx, y' = -sinx.
Applications
Study Notes
Introduction
These notes systematically organize trigonometric knowledge for
Calculus I (Math 150A), bridging algebraic manipulation with
geometric intuition and real-world applications. The focus is on
mastering trigonometric identities and properties for calculus
problem-solving, emphasizing critical insights beyond basic formula
memorization.
Core Trigonometric Identities
Core Idea: Mastering trigonometric identities is essential for
simplification, differentiation, and integration of trigonometric
functions in calculus.
Pythagorean Identities: Fundamental identities relating squares
of sine, cosine, and tangent.
sin²θ + cos²θ = 1, tan θ = sinθ/cosθ.
Key Derivations: Useful for simplifying integrals and derivatives.
sin²θ = 1 - cos²θ, cos²θ = 1 - sin²θ.
Double-Angle Formulas: Express trigonometric functions of 2θ in
terms of θ.
sin2θ = 2sinθcosθ, cos2θ = cos²θ – sin²θ = 2cos²θ − 1, tan2θ =
2tanθ / (1 - tan²θ).
Power-Reduction Formulas: Useful for integrating sin²x or cos²x.
sin²θ = (1 - cos2θ)/2, cos²θ = (1 + cos2θ)/2.
Sum and Difference Formulas: Essential for evaluating
trigonometric functions of non-special angles.
cos(α + β) = cosαcosβ – sinαsinβ, cos(α – β) = cosαcosβ + sinαsinβ.
sin(α + β) = sinαcosβ + cosαsinβ, sin(α – β) = sinαcosβ – cosαsinβ.
Example 1.1: Simplifying Using Pythagorean Identity
, Problem: Simplify the expression
1− cos2 x
sinx
Solution:
By the Pythagorean identity, 1 −cos 2 x=sin 2 x .
Thus,
sin 2 x
=sinx
sinx
Key Terminology:
Pythagorean identity: A fundamental trigonometric identity
derived from the unit circle.
Simplification: Reducing expressions to their most compact
and interpretable form.
Properties of Sine and Cosine Functions
Core Idea: Understanding the properties of sine and cosine
functions is crucial for calculus analysis, including analyzing function
behavior and solving related problems.
Core Properties Summary: Key properties aligned with calculus
applications.
Domain: R (all real numbers) for both sinx and cosx.
Range: [-1, 1] for both sinx and cosx.
Periodicity: Minimum positive period of 2π for both sinx and cosx.
Parity: sin(–x) = –sinx (odd function), cos(–x) = cosx (even function).
Monotonic Intervals: Intervals of increasing and decreasing behavior.
Extrema: Maximum and minimum values and their locations.
Calculus-Focused Interpretations: Periodicity and Integration
Integrals over intervals of length 2π can be simplified due to
periodicity.
Monotonicity and Derivatives: The monotonic intervals relate
directly to the sign of the derivative.
For y = sinx, y' = cosx.
For y = cosx, y' = -sinx.
