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 5.12aLial - 12th Edition
Textbook Question
Find the exact value of each expression. (Do not use a calculator.)
cos 105° (Hint: 105° = 60° + 45°)
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Recognize that the problem involves finding the cosine of an angle that can be expressed as the sum of two known angles: 105° = 60° + 45°.
Use the cosine addition formula: .
Substitute and into the formula: .
Recall the exact trigonometric values: , , , and .
Substitute these values into the expression: .
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Cosine Addition Formula
The cosine addition formula states that cos(A + B) = cos(A)cos(B) - sin(A)sin(B). This formula allows us to find the cosine of an angle that is the sum of two other angles, which is essential for calculating cos(105°) as 105° can be expressed as 60° + 45°.
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Special Angles in Trigonometry
In trigonometry, certain angles like 30°, 45°, and 60° have known sine and cosine values. For example, cos(60°) = 1/2 and cos(45°) = √2/2. Recognizing these special angles helps simplify calculations and find exact values without a calculator.
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Quadrants and Angle Signs
Understanding the unit circle and the signs of trigonometric functions in different quadrants is crucial. Since 105° is in the second quadrant, where cosine values are negative, this knowledge is important when determining the final value of cos(105°) after applying the cosine addition formula.
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Related Practice
Textbook Question
Be sure that you've familiarized yourself with the first set of formulas presented in this section by working C1–C4 in the Concept and Vocabulary Check. In Exercises 1–8, use the appropriate formula to express each product as a sum or difference.
sin x cos 2x
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