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Understanding Trigonometric Functions Using the Unit Circle

This guide explores the importance of the unit circle in identifying and evaluating trigonometric functions. Each real number wraps around the unit circle and corresponds to a point representing a central angle in standard position. Learn how to determine the coordinates based on this relationship and evaluate the six trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent—using both the unit circle and a calculator. Ensure your calculator is set to the correct mode (degrees or radians) for accurate results, and round answers to four decimal places.

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Understanding Trigonometric Functions Using the Unit Circle

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  1. Trigonometric Functions: The Unit Circle Objectives: Identify a unit circle Evaluate trigonometric functions using the unit circle and a calculator

  2. The Unit Circle

  3. The Unit Circle • As the real number line wraps around the unit circle, each real number corresponds to a point on the circle. For example, the real number corresponds to the point . • In general, each real number corresponds to a central angle (in standard position) whose radian measure is • It follows that the coordinates are two functions of the real variable .

  4. Definitions of Trigonometric functions • Let be a real number and let be the point on the unit circle corresponding to .

  5. EX: Evaluate the six trigonometric functions at each real number • 1. • 2.

  6. EX: Evaluate the six trigonometric functions at each real number • 3. • 4.

  7. EX: Evaluate the six trigonometric functions at each real number • 5. • 6.

  8. EX: Evaluate the six trigonometric functions at each real number • 7. • 8.

  9. Evaluating trigonometric function • When evaluating a trigonometric function with a calculator, you need to set the calculator to the desired mode of measurement (degree or radian). Most calculators do not have keys for cosecant, secant, and cotangent functions. To evaluate these functions, you can use the reciprocal key with their respective reciprocal functions: sine, cosine and tangent. Round the answers to four decimal places.

  10. EX: Use a calculator to evaluate the trigonometric function. Degrees Radians • 9. • 10. • 11. • 12. • 13. • 14.

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