AP Precalculus Section 3.12 Example: Trigonometric Angle Addition and Subtraction Property

Random AP Precalculus Problems (I found on the Internet). These are not official AP Collegeboard examples, but they will definitely get the job done!

The Trigonometric Angle Addition and Subtraction Formulas express the sine and cosine of the sum or difference of two angles in terms of the sines and cosines of the individual angles. These formulas are useful for simplifying trigonometric expressions and solving trigonometric equations.

### Sine Angle Addition and Subtraction Formula:
\[ \sin(A \pm B) = \sin(A)\cos(B) \pm \cos(A)\sin(B) \]

### Cosine Angle Addition and Subtraction Formula:
\[ \cos(A \pm B) = \cos(A)\cos(B) \mp \sin(A)\sin(B) \]

In these formulas:
- \( A \) and \( B \) are angles.
- \( \sin(A \pm B) \) represents the sine of the sum or difference of \( A \) and \( B \).
- \( \cos(A \pm B) \) represents the cosine of the sum or difference of \( A \) and \( B \).
- The \( \pm \) sign in the formulas indicates that there are two possible formulas: one with addition and one with subtraction.

### Examples:

1. **Sine Angle Addition:**
\[ \sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B) \]

2. **Sine Angle Subtraction:**
\[ \sin(A - B) = \sin(A)\cos(B) - \cos(A)\sin(B) \]

3. **Cosine Angle Addition:**
\[ \cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B) \]

4. **Cosine Angle Subtraction:**
\[ \cos(A - B) = \cos(A)\cos(B) + \sin(A)\sin(B) \]

### Use Cases:

1. **Solving Trigonometric Equations:**
These formulas are often used to simplify trigonometric expressions and solve equations involving sums or differences of angles.

2. **Geometry and Physics:**
In geometry and physics, these formulas are useful for analyzing the components of vectors or forces in different directions.

3. **Trigonometric Identities:**
The angle addition and subtraction formulas are fundamental in establishing various trigonometric identities.

The Topics covered in AP Precalculus are...

1.1 Change in Tandem
1.2 Rates of Change
1.3 Rates of Change in Linear and Quadratic Functions
1.4 Polynomial Functions and Rates of Change
1.5 Polynomial Functions and Complex Zeros
1.6 Polynomial Functions and End Behavior
1.7 Rational Functions and End Behavior
1.8 Rational Functions and Zeros
1.9 Rational Functions and Vertical Asymptotes
1.10 Rational Functions and Holes
1.11 Equivalent Representations of Polynomial and Rational Expressions
1.12 Transformations of Functions
1.13 Function Model Selection and Assumption Articulation
1.14 Function Model Construction and Application
2.1 Change in Arithmetic and Geometric Sequences
2.2 Change in Linear and Exponential Functions
2.3 Exponential Functions
2.4 Exponential Function Manipulation
2.5 Exponential Function Context and Data Modeling
2.6 Competing Function Model Validation
2.7 Composition of Functions
2.8 Inverse Functions
2.9 Logarithmic Expressions
2.10 Inverses of Exponential Functions
2.11 Logarithmic Functions
2.12 Logarithmic Function Manipulation
2.13 Exponential and Logarithmic Equations and Inequalities
2.14 Logarithmic Function Context and Data Modeling
2.15 Semi-log Plots
3.1 Periodic Phenomena
3.2 Sine, Cosine, and Tangent
3.3 Sine and Cosine Function Values
3.4 Sine and Cosine Function Graphs
3.5 Sinusoidal Functions
3.6 Sinusoidal Function Transformations
3.7 Sinusoidal Function Context and Data Modeling
3.8 The Tangent Function
3.9 Inverse Trigonometric Functions
3.10 Trigonometric Equations and Inequalities
3.11 The Secant, Cosecant, and Cotangent Functions
3.12 Equivalent Representations of Trigonometric Functions
3.13 Trigonometry and Polar Coordinates
3.14 Polar Function Graphs
3.15 Rates of Change in Polar Functions

I have many informative videos for Pre-Algebra, Algebra 1, Algebra 2, Geometry, Pre-Calculus, and Calculus. Please check it out:

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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Hi, can you also please post a review video on the rest of unit 3? Up to 3.15