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7.2.1 – Sum and Difference Identities

In this lesson, we explore the Sum and Difference Identities, crucial for calculating exact trigonometric values of unknown angles on the unit circle. These identities enhance precision in engineering and anatomy by allowing the computation of exact angle measures, such as flexion angles. We introduce formulas for sine, cosine, and tangent of sums and differences of angles, focusing on practical applications. This approach encourages avoiding calculators and instead relies on fundamental angle relationships. Through examples and assignments, students gain proficiency in evaluating trigonometric functions accurately.

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7.2.1 – Sum and Difference Identities

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  1. 7.2.1 – Sum and Difference Identities

  2. A new set of identities we will deal with will allow us to determine exact trig values of angles on the unit circle that we do not know • Example: sin(350)?

  3. Motivation allows us to avoid calculator use, and be more precise when talking of applied terms • Engineering: Must know exact angles to measure • Anatomy: Exact values when talking of angles of flexion or similar

  4. Sum and Diff: Sine • Let u and v be two unique angles • sin(u + v) = sin(u)cos(v) + sin(v)cos(u) • sin(u – v) = sin(u)cos(v) – sin(v)cos(u)

  5. Sum and Diff: Cos • Let u and v be two unique angles • cos(u + v) = cos(u)cos(v) - sin(v)sin(u) • cos(u – v) = cos(u)cos(v) + sin(v)sin(u)

  6. Sum and Diff: Tan • Let u and v be two unique angles:

  7. Using these, we can determine angles in two ways: • 1) Use literally in the sense of u + v • 2) Write an angle as the sum of two known angles from the unit circle (30, 45, 60, 90,…) • Always make sure to use angles whose sin, cos, or tan values can be readily referenced (go back to our chart)

  8. Example. Determine the value of • Cos( )

  9. Example. Determine the value of sin( )

  10. Example. Determine the exact value of tan( )

  11. If we need to find the value of an angle, say for 1950, we must determine what sum, or difference, of two angles we can reference from the unit circle that we may use

  12. Example. Determine the exact value of cos(1950)

  13. Example. Determine the exact value of cos( )

  14. Assignment • Pg. 564 • 1, 2, 4, 13, 15, 21, 24, 27, 39

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