AP Precalculus Section 3.3 Example: Vertical Distance Between Two Points on a Circle with Radius 16

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

To find the vertical distance between two points on a circle with radius 16 and express the solution in terms of sine, you can follow these steps:

1. **Understand the Coordinates:**
- Let the two points on the circle be \(P_1(x_1, y_1)\) and \(P_2(x_2, y_2)\).

2. **Vertical Distance Formula:**
- The vertical distance (\(\Delta y\)) between two points is given by the absolute difference of their y-coordinates: \(\Delta y = |y_2 - y_1|\).

3. **Use Trigonometric Definitions:**
- Recognize that the y-coordinate on the circle with radius 16 is given by \(y = r \sin \theta\), where \(r\) is the radius and \(\theta\) is the angle.

4. **Express y in Terms of Sine:**
- Express the y-coordinates of \(P_1\) and \(P_2\) in terms of sine:
- \(y_1 = 16 \sin \theta_1\)
- \(y_2 = 16 \sin \theta_2\)

5. **Calculate Vertical Distance in Terms of Sine:**
- Substitute these expressions into the vertical distance formula:
- \(\Delta y = |16 \sin \theta_2 - 16 \sin \theta_1|\)

6. **Example:**
- If \(P_1\) corresponds to \(\theta_1 = \frac{\pi}{4}\) and \(P_2\) corresponds to \(\theta_2 = \frac{3\pi}{4}\), then
- \(\Delta y = |16 \sin \frac{3\pi}{4} - 16 \sin \frac{\pi}{4}|\)

7. **Simplify as Needed:**
- Use trigonometric identities to simplify the expression further if possible.

In summary, to find the vertical distance between two points on a circle with radius 16 in terms of sine, express the y-coordinates in terms of sine and apply the vertical distance formula.

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

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

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