Table of contents
- 0. Review of College Algebra4h 43m
- 1. Measuring Angles39m
- 2. Trigonometric Functions on Right Triangles2h 5m
- 3. Unit Circle1h 19m
- 4. Graphing Trigonometric Functions1h 19m
- 5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
- 6. Trigonometric Identities and More Equations2h 34m
- 7. Non-Right Triangles1h 38m
- 8. Vectors2h 25m
- 9. Polar Equations2h 5m
- 10. Parametric Equations1h 6m
- 11. Graphing Complex Numbers1h 7m
6. Trigonometric Identities and More Equations
Sum and Difference Identities
Problem 5.56
Textbook Question
Textbook QuestionFind cos(s + t) and cos(s - t).
cos s = √2/4 and sin t = - √5/6, s and t in quadrant IV
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Cosine Addition and Subtraction Formulas
The cosine addition and subtraction formulas are essential for finding the cosine of the sum or difference of two angles. Specifically, cos(s + t) = cos s * cos t - sin s * sin t and cos(s - t) = cos s * cos t + sin s * sin t. These formulas allow us to express the cosine of combined angles in terms of the cosines and sines of the individual angles.
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Quadrant IV Trigonometric Values
In Quadrant IV, the cosine values are positive while the sine values are negative. Given that cos s = √2/4 is positive, we can directly use this value. However, since sin t = -√5/6 is negative, we need to determine cos t using the Pythagorean identity, which states that sin² t + cos² t = 1, to find the cosine of angle t.
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Pythagorean Identity
The Pythagorean identity is a fundamental relationship in trigonometry that connects the sine and cosine of an angle. It states that sin² θ + cos² θ = 1 for any angle θ. This identity is particularly useful for finding missing trigonometric values when one is known, allowing us to calculate cos t from sin t in this problem.
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