Course Work: Trigonometric Functions in Calculus I (Math 150)
Institution: California State University, Northridge (Northridge, CA)
Course: Math 150 (Calculus I)
Date: November
Instructor:David Klein
1. Introduction
In Math 150 (Calculus I), trigonometric functions transcend their pre-calculus role
as "angle-based tools" to become essential for analyzing continuous change,
area/volume calculations, and real-world systems (e.g., satellite orbits, periodic
motion). Unlike basic trigonometry courses, Math 150 requires students to:
1. Use trigonometric identities to simplify calculus operations (e.g., reducing
powers of sine/cosine for integration).
2. Apply calculus to analyze trigonometric function behavior (e.g., finding
extrema via derivatives).
3. Model real-world problems with trigonometric functions and solve them
using calculus.
2. Core Trigonometric Identities for Calculus
Trigonometric identities are the "language" that translates complex trigonometric
expressions into forms manageable for calculus. Below are the key identities, their
derivations, and their specific uses in Math 150.
2.1 Pythagorean Identities
The Pythagorean identities arise from the unit circle definition of sine and cosine,
where sin� and cos� represent the �- and �-coordinates of a point on the unit circle
(�2 + �2 = 1).
Key Identities:
sin�
sin2 � + cos2 � = 1tan� = (cos� ≠ 0)
cos�
Derived Forms (Critical for Calculus):
By rearranging equation (1), we get:
, sin2 � = 1 − cos2 � (1�)cos2 � = 1 − sin2 � (1�)
Math 150 Application: Simplifying Integrals
A common Math 150 integral is ∫sin�cos2 ���. Using identity (1b), we can substitute
cos2 � = 1 − sin2 �, but a more efficient approach uses substitution: let � = cos�, �� =
�3 cos3 �
− sin���, so the integral becomes −∫�2 �� =− 3 + � =− 3 + �. Without
recognizing the Pythagorean relationship, this substitution would be far less
intuitive.
2.2 Double-Angle Formulas
Double-angle formulas express sin2�, cos2�, and tan2� in terms of �, enabling the
reduction of "higher-degree" trigonometric terms to linear ones—essential for
integration.
Key Identities:
sin2� = 2sin�cos�cos2� = cos2 � − sin2 �cos2� = 2cos2 � − 1 (from (1) and (4))cos2� = 1 − 2sin2�
Power-Reduction Formulas (Math 150 Staple):
By rearranging (5) and (6), we derive power-reduction formulas—critical for
integrating sin2 � or cos2 � (which cannot be integrated directly using basic power
rules):
1 − cos2� 1 + cos2�
sin2 � = (7)cos2 � = (8)
2 2
�
Example: Integrate ∫0 sin2 ���
Using formula (7):
� � � �
1 − cos2� 1 1
sin2 ��� = �� = 1 �� − cos 2���
0 0 2 2 0 2 0
Evaluate term-by-term:
1 � 1 �
∫ 1 ��
2 0
= (� − 0) =
2 2
1 � 1 sin2� � 1
∫ cos 2��� = 2
2 0 2
= 4 (sin2� − sin0) = 0
0
�
Thus, the integral equals 2—a result that would be impossible without power-
reduction.
Institution: California State University, Northridge (Northridge, CA)
Course: Math 150 (Calculus I)
Date: November
Instructor:David Klein
1. Introduction
In Math 150 (Calculus I), trigonometric functions transcend their pre-calculus role
as "angle-based tools" to become essential for analyzing continuous change,
area/volume calculations, and real-world systems (e.g., satellite orbits, periodic
motion). Unlike basic trigonometry courses, Math 150 requires students to:
1. Use trigonometric identities to simplify calculus operations (e.g., reducing
powers of sine/cosine for integration).
2. Apply calculus to analyze trigonometric function behavior (e.g., finding
extrema via derivatives).
3. Model real-world problems with trigonometric functions and solve them
using calculus.
2. Core Trigonometric Identities for Calculus
Trigonometric identities are the "language" that translates complex trigonometric
expressions into forms manageable for calculus. Below are the key identities, their
derivations, and their specific uses in Math 150.
2.1 Pythagorean Identities
The Pythagorean identities arise from the unit circle definition of sine and cosine,
where sin� and cos� represent the �- and �-coordinates of a point on the unit circle
(�2 + �2 = 1).
Key Identities:
sin�
sin2 � + cos2 � = 1tan� = (cos� ≠ 0)
cos�
Derived Forms (Critical for Calculus):
By rearranging equation (1), we get:
, sin2 � = 1 − cos2 � (1�)cos2 � = 1 − sin2 � (1�)
Math 150 Application: Simplifying Integrals
A common Math 150 integral is ∫sin�cos2 ���. Using identity (1b), we can substitute
cos2 � = 1 − sin2 �, but a more efficient approach uses substitution: let � = cos�, �� =
�3 cos3 �
− sin���, so the integral becomes −∫�2 �� =− 3 + � =− 3 + �. Without
recognizing the Pythagorean relationship, this substitution would be far less
intuitive.
2.2 Double-Angle Formulas
Double-angle formulas express sin2�, cos2�, and tan2� in terms of �, enabling the
reduction of "higher-degree" trigonometric terms to linear ones—essential for
integration.
Key Identities:
sin2� = 2sin�cos�cos2� = cos2 � − sin2 �cos2� = 2cos2 � − 1 (from (1) and (4))cos2� = 1 − 2sin2�
Power-Reduction Formulas (Math 150 Staple):
By rearranging (5) and (6), we derive power-reduction formulas—critical for
integrating sin2 � or cos2 � (which cannot be integrated directly using basic power
rules):
1 − cos2� 1 + cos2�
sin2 � = (7)cos2 � = (8)
2 2
�
Example: Integrate ∫0 sin2 ���
Using formula (7):
� � � �
1 − cos2� 1 1
sin2 ��� = �� = 1 �� − cos 2���
0 0 2 2 0 2 0
Evaluate term-by-term:
1 � 1 �
∫ 1 ��
2 0
= (� − 0) =
2 2
1 � 1 sin2� � 1
∫ cos 2��� = 2
2 0 2
= 4 (sin2� − sin0) = 0
0
�
Thus, the integral equals 2—a result that would be impossible without power-
reduction.
