Math 150 (Calculus I) Study Notes: Trigonometric Functions &
Applications
Institution: California State University, Northridge (Northridge, CA)
Course: Math 150 (Calculus I)
Date: November
Preface: Learning Insights
Trigonometric functions are foundational in Calculus I, bridging algebraic
manipulation with geometric intuition and real-world applications. Through
working through college entrance exam problems (adapted to align with Math 150
curriculum), I’ve gained key insights that extend beyond basic formula
memorization:
First, contextual formula selection is critical. Unlike simple textbook exercises,
real problems (e.g., satellite coverage, function value evaluation) require identifying
which identity (Pythagorean, double-angle, sum/difference) fits the given
conditions—often combining multiple formulas in sequence. For example, solving
for a trigonometric value may first use the Pythagorean identity to find a missing
term, then apply the sum/difference formula to compute the target.
Second, trigonometric functions’ properties drive application solving.
Understanding periodicity, monotonicity, and range isn’t just theoretical; it’s
essential for analyzing function behavior (e.g., finding monotonic intervals) and
solving optimization or real-world modeling problems (e.g., satellite signal coverage
area).
Third, geometric interpretation enhances problem-solving. Many trigonometric
problems (like the satellite orbit question) have underlying geometric meanings
(angles, spherical surfaces). Translating algebraic expressions to geometric concepts
helps simplify complex calculations and verify results.
This note systematically organizes trigonometric knowledge covered in the practice
problems, integrates Math 150’s calculus-focused perspective, and includes detailed
examples to reinforce understanding and application.
, Chapter 1: Core Trigonometric Identities (Foundational for Calculus)
Trigonometric identities are the building blocks for simplification, differentiation,
and integration of trigonometric functions in Math 150. Mastering these identities is
essential for tackling calculus problems involving trigonometric expressions.
1.1 Pythagorean Identities
The most fundamental identities relate the squares of sine and cosine, and their
ratio to tangent:
sin�
sin2 � + cos2 � = 1tan� = (cos� ≠ 0)
cos�
Key Derivations (Useful in Calculus):
From equation (1), we can derive two additional forms by rearranging terms—
critical for simplifying integrals and derivatives:
sin2 � = 1 − cos2 �cos2 � = 1 − sin2 �
Math 150 Insight: These identities are frequently used to simplify expressions
before differentiation (e.g., reducing powers of sine/cosine) or integration (e.g.,
substituting 1 − sin2 � for cos2� to use the power rule).
1.2 Double-Angle Formulas
Double-angle formulas express trigonometric functions of 2� in terms of �, a staple
in calculus for simplifying complex expressions and solving differential equations:
sin2� = 2sin�cos�cos2� = cos2 � − sin2 �cos2� = 2cos2 � − 1 ved from (1) and (4))�
2tan�
(derived from (1) and (4))tan2� = (tan� ≠± 1)
1 − tan2 �
Math 150 Insight: Formulas (5) and (6) are particularly useful for integrating sin2 �
or cos2 � (a common Math 150 integral problem). By rearranging them, we get the
"power-reduction formulas":
1 − cos2� 1 + cos2�
sin2 � = , cos2 � =
2 2
These convert squared terms to linear terms in cos2�, making integration
straightforward.
1.3 Sum and Difference Formulas
These formulas are essential for evaluating trigonometric functions of non-special
angles and simplifying composite trigonometric expressions—critical for calculus
problems involving function composition:
Applications
Institution: California State University, Northridge (Northridge, CA)
Course: Math 150 (Calculus I)
Date: November
Preface: Learning Insights
Trigonometric functions are foundational in Calculus I, bridging algebraic
manipulation with geometric intuition and real-world applications. Through
working through college entrance exam problems (adapted to align with Math 150
curriculum), I’ve gained key insights that extend beyond basic formula
memorization:
First, contextual formula selection is critical. Unlike simple textbook exercises,
real problems (e.g., satellite coverage, function value evaluation) require identifying
which identity (Pythagorean, double-angle, sum/difference) fits the given
conditions—often combining multiple formulas in sequence. For example, solving
for a trigonometric value may first use the Pythagorean identity to find a missing
term, then apply the sum/difference formula to compute the target.
Second, trigonometric functions’ properties drive application solving.
Understanding periodicity, monotonicity, and range isn’t just theoretical; it’s
essential for analyzing function behavior (e.g., finding monotonic intervals) and
solving optimization or real-world modeling problems (e.g., satellite signal coverage
area).
Third, geometric interpretation enhances problem-solving. Many trigonometric
problems (like the satellite orbit question) have underlying geometric meanings
(angles, spherical surfaces). Translating algebraic expressions to geometric concepts
helps simplify complex calculations and verify results.
This note systematically organizes trigonometric knowledge covered in the practice
problems, integrates Math 150’s calculus-focused perspective, and includes detailed
examples to reinforce understanding and application.
, Chapter 1: Core Trigonometric Identities (Foundational for Calculus)
Trigonometric identities are the building blocks for simplification, differentiation,
and integration of trigonometric functions in Math 150. Mastering these identities is
essential for tackling calculus problems involving trigonometric expressions.
1.1 Pythagorean Identities
The most fundamental identities relate the squares of sine and cosine, and their
ratio to tangent:
sin�
sin2 � + cos2 � = 1tan� = (cos� ≠ 0)
cos�
Key Derivations (Useful in Calculus):
From equation (1), we can derive two additional forms by rearranging terms—
critical for simplifying integrals and derivatives:
sin2 � = 1 − cos2 �cos2 � = 1 − sin2 �
Math 150 Insight: These identities are frequently used to simplify expressions
before differentiation (e.g., reducing powers of sine/cosine) or integration (e.g.,
substituting 1 − sin2 � for cos2� to use the power rule).
1.2 Double-Angle Formulas
Double-angle formulas express trigonometric functions of 2� in terms of �, a staple
in calculus for simplifying complex expressions and solving differential equations:
sin2� = 2sin�cos�cos2� = cos2 � − sin2 �cos2� = 2cos2 � − 1 ved from (1) and (4))�
2tan�
(derived from (1) and (4))tan2� = (tan� ≠± 1)
1 − tan2 �
Math 150 Insight: Formulas (5) and (6) are particularly useful for integrating sin2 �
or cos2 � (a common Math 150 integral problem). By rearranging them, we get the
"power-reduction formulas":
1 − cos2� 1 + cos2�
sin2 � = , cos2 � =
2 2
These convert squared terms to linear terms in cos2�, making integration
straightforward.
1.3 Sum and Difference Formulas
These formulas are essential for evaluating trigonometric functions of non-special
angles and simplifying composite trigonometric expressions—critical for calculus
problems involving function composition:
