Trigonometric Functions: Study Notes
Course: Math 150A (Calculus I)
Institution: California State University, Northridge
Instructor: David Klein
Date: November
Part I: Basic Concepts (Basic Level)
Definition: Trigonometric functions (sine, cosine, tangent, cotangent, secant,
cosecant) relate angles of a right triangle to ratios of its sides.
Key Properties:
oDomain: The set of all possible input values (angles).
oRange: The set of all possible output values.
oPeriodicity: Functions repeat their values at regular intervals.
Basic Identities:
osin² + cos² = 1θ θ
otan = sin / cos θ θ θ
ocot = 1 / tan θ θ
osec = 1 / cos θ θ
ocsc = 1 / sin θ θ
Part II: Graphs and Properties (Intermediate Level)
Periodicity:
oSine and Cosine: Period = 2π
oTangent and Cotangent: Period = π
Monotonicity: Intervals where the function is increasing or decreasing.
oExample: Sine is increasing on [- /2, /2] and decreasing on [ /2, π π π
3 /2].π
Parity:
oEven Functions: f(-x) = f(x) (Cosine)
oOdd Functions: f(-x) = -f(x) (Sine, Tangent)
Transformations:
oAmplitude: Affects the maximum and minimum values.
oPeriod Change: Horizontal stretch or compression.
oPhase Shift: Horizontal translation.
oVertical Shift: Vertical translation.
Domain and Range:
oSine and Cosine: Domain = (-∞, ∞), Range = [-1, 1]
oTangent: Domain = x ≠ ( /2) + n , Range = (-∞, ∞)π π
Part III: Identities and Transformations (Intermediate-Advanced Level)
Sum and Difference Formulas:
osin(A ± B) = sin A cos B ± cos A sin B
ocos(A ± B) = cos A cos B sin A sin B∓
otan(A ± B) = (tan A ± tan B) / (1 tan A tan B)∓
Double Angle Formulas:
osin 2 = 2 sin cos θ θ θ
ocos 2 = cos² - sin² = 2 cos² - 1 = 1 - 2 sin²θ θ θ θ θ
otan 2 = (2 tan ) / (1 - tan² )θ θ θ
Half Angle Formulas:
osin ( /2) = ±√((1 - cos ) / 2)θ θ
ocos ( /2) = ±√((1 + cos ) / 2)θ θ
otan ( /2) = ±√((1 - cos ) / (1 + cos )) = sin / (1 + cos ) = (1 - cos θ θ θ θ θ
) / sin θ θ
Product-to-Sum Formulas:
osin A cos B = 1/2 [sin(A + B) + sin(A - B)]
ocos A sin B = 1/2 [sin(A + B) - sin(A - B)]
ocos A cos B = 1/2 [cos(A + B) + cos(A - B)]
osin A sin B = -1/2 [cos(A + B) - cos(A - B)]
Sum-to-Product Formulas:
osin A + sin B = 2 sin((A + B) / 2) cos((A - B) / 2)
osin A - sin B = 2 cos((A + B) / 2) sin((A - B) / 2)
ocos A + cos B = 2 cos((A + B) / 2) cos((A - B) / 2)
ocos A - cos B = -2 sin((A + B) / 2) sin((A - B) / 2)
Part IV: Solving Triangles (Intermediate-Advanced Level)
Law of Sines: a / sin A = b / sin B = c / sin C
oUsed to solve triangles when given two angles and a side (AAS or ASA)
or two sides and an angle opposite one of them (SSA).
Law of Cosines:
oa² = b² + c² - 2bc cos A
ob² = a² + c² - 2ac cos B
oc² = a² + b² - 2ab cos C
oUsed to solve triangles when given three sides (SSS) or two sides and
the included angle (SAS).
Area Formulas:
oArea = (1/2) * b * c * sin A
oArea = √[s(s - a)(s - b)(s - c)] (Herons formula, where s is the semi-
perimeter)
Part V: Trigonometric Functions and Vectors (Advanced Level)
Vector Representation: Representing points and directions using
trigonometric functions.
Dot Product: a · b = a b cos θ
Magnitude: a = √(x² + y²) where a = (x, y)
Angle Between Vectors: cos = (θa · b) / (a b)
Applications: Coordinate geometry problems involving angles and
distances.
Part VI: Comprehensive Applications (Advanced Level)
Trigonometry and Calculus: Using trigonometric functions in
differentiation and integration.
Analytic Geometry: Combining trigonometric functions with geometric
shapes and equations.
Real-World Applications: Modeling periodic phenomena, such as
oscillations and waves.
Problem-Solving Strategies: Combining multiple concepts to solve complex
problems.
Study Guidelines and Summary
Learning Method Recommendations:
oBasic Concepts: Memorize definitions, properties, and relationships.
oGraphs and Properties: Understand transformations and key
features.
oIdentities and Transformations: Practice applying formulas and
simplifying expressions.
oSolving Triangles: Master the Law of Sines and Law of Cosines.
oComprehensive Applications: Develop problem-solving skills and
mathematical modeling abilities.
Practice Strategies:
oProgressive Learning: Start with basic problems and gradually
increase complexity.
oSolution Analysis: Carefully study the solution process for each
problem.
oStep-by-Step Approach: Break down complex problems into
manageable steps.
oRegular Review: Periodically review concepts and formulas.
oPattern Recognition: Identify common problem types and solution
methods.
oApplication Focus: Practice applying concepts to real-world
scenarios.
Common Problem-Solving Techniques:
oIdentity Verification: Use fundamental identities to prove or
disprove equations.
oAngle Manipulation: Convert between different angle measures and
relationships.
oGraph Analysis: Use graphical properties to solve equations and
inequalities.
oSubstitution Methods: Replace complex expressions with simpler
equivalent forms.
oGeometric Interpretation: Visualize problems using unit circles and
triangles.
Tips for Success:
oMemorize Key Formulas
oPractice Regularly
oUnderstand Concepts
oUse Multiple Approaches
oCheck Your Work
Essential Formula Reference
Basic Identities: sin² + cos² = 1, tan = sin / cos , etc.θ θ θ θ θ
Sum and Difference Formulas: sin(A ± B), cos(A ± B), tan(A ± B)
Double Angle Formulas: sin 2 , cos 2 , tan 2θ θ θ
Half Angle Formulas: sin ( /2), cos ( /2), tan ( /2)θ θ θ
Law of Sines: a / sin A = b / sin B = c / sin C
Law of Cosines: a² = b² + c² - 2bc cos A
Triangle Area: Area = (1/2) * b * c * sin A
