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
3:51 minutes
Problem 5b
Textbook Question
Textbook QuestionUse 105° = 135° - 30° to find the exact value of 105°.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Angle Addition and Subtraction
The angle addition and subtraction formulas are fundamental in trigonometry, allowing us to express the sine, cosine, and tangent of sums or differences of angles. For example, sin(a ± b) = sin(a)cos(b) ± cos(a)sin(b) helps in calculating the trigonometric values of angles that are not standard. Understanding these formulas is essential for breaking down complex angles into manageable components.
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Reference Angles
Reference angles are the acute angles formed by the terminal side of an angle and the x-axis. They are crucial for determining the sine, cosine, and tangent values of angles in different quadrants. For instance, the reference angle for 105° is 75°, which helps in finding the exact trigonometric values by relating them to known angles.
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Exact Values of Trigonometric Functions
Exact values of trigonometric functions refer to the precise values of sine, cosine, and tangent for specific angles, often derived from special triangles or the unit circle. For example, knowing that sin(30°) = 1/2 and cos(30°) = √3/2 allows us to compute values for angles like 105° using angle addition or subtraction. Mastery of these exact values is essential for solving trigonometric equations and problems.
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