AP Precalculus Section 3.4 Example: Sine of an Angle on the Unit Circle

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

If you are given a coordinate on the unit circle and want to find the sine of the corresponding angle, you can follow these steps:

1. **Understand the Coordinates:**
- The unit circle is centered at the origin (0,0) with a radius of 1. Given a point (x, y) on the unit circle, the coordinates represent \((\cos \theta, \sin \theta)\), where \(\theta\) is the angle formed by the radius and the positive x-axis.

2. **Identify the Sine:**
- The y-coordinate of the point on the unit circle is equal to the sine of the angle: \(\sin \theta = y\).

3. **Example:**
- If you have a point (0.6, 0.8) on the unit circle, the sine of the corresponding angle is \(\sin \theta = 0.8\).

4. **Verify with Pythagorean Identity:**
- You can verify your result using the Pythagorean Identity: \(\sin^2 \theta + \cos^2 \theta = 1\). In the example above, check if \(0.8^2 + 0.6^2 = 1\), which should hold true for points on the unit circle.

In summary, to find the sine of an angle on the unit circle given a coordinate, simply use the y-coordinate of the point. The sine of the angle is equal to the y-coordinate, and you can verify the result using the Pythagorean Identity.

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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