AP Precalculus Section 3.3 Example: Sine and Cosine on a Unit Circle (Coordinates)

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 sine (\(\sin A\)) and cosine (\(\cos A\)) of an angle \(A\) given a coordinate on the unit circle, follow these steps:

1. **Understand the Coordinate System:**
- The unit circle is centered at the origin (0,0) and has a radius of 1. Coordinates on the unit circle are given in the form (x, y).

2. **Given Coordinate:**
- If you have a coordinate (x, y) on the unit circle corresponding to an angle \(A\), that point represents where the terminal side of the angle intersects the unit circle.

3. **Cosine (\(\cos A\)):**
- The x-coordinate of the point is equal to \(\cos A\). In mathematical terms: \(\cos A = x\).

4. **Sine (\(\sin A\)):**
- The y-coordinate of the point is equal to \(\sin A\). In mathematical terms: \(\sin A = y\).

5. **Example:**
- Let's say you have a point on the unit circle with coordinates (0.6, 0.8). If this corresponds to an angle \(A\), then:
- \(\cos A = 0.6\)
- \(\sin A = 0.8\)

6. **Verify with Pythagorean Identity:**
- You can also verify your results using the Pythagorean Identity: \(\sin^2 A + \cos^2 A = 1\). Substitute the values of \(\sin A\) and \(\cos A\) into this equation. In the example above, you would have \(0.8^2 + 0.6^2 = 1\), which confirms the point is on the unit circle.

In summary, to find the sine and cosine of an angle given a coordinate on the unit circle, use the x-coordinate for cosine and the y-coordinate for sine.

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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