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14-5 Sum and Difference of Angles Formulas

14-5 Sum and Difference of Angles Formulas. The Formulas. Use the formula . Sum of angles. Evaluate each expression. Example 5-1a. Example:. Find the exact value of sin 75 . Multiply. Simplify. Answer:. Example 5-1b. Use the formula. Difference of angles.

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14-5 Sum and Difference of Angles Formulas

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  1. 14-5 Sum and Difference of Angles Formulas

  2. The Formulas

  3. Use the formula . Sum of angles Evaluateeach expression. Example 5-1a Example: Find the exact value of sin 75.

  4. Multiply. Simplify. Answer: Example 5-1b

  5. Use the formula Difference of angles Evaluate eachexpression. Example 5-1c Another one: Find the exact value of cos (–75).

  6. Multiply. Simplify. Answer: Example 5-1d

  7. Answer: Answer: Example 5-1e More: Find the exact value of each expression. a. sin 105 b. cos (–120 )

  8. Verify that is an identity. Answer: Original equation Difference ofangles formula Evaluate eachexpression. Simplify. Example 5-3a Another one:

  9. Verify that is an identity. Original equation Answer: Difference ofangles formula Evaluate eachexpression. Simplify. Example 5-3b Another one:

  10. Verify that each of the following is an identity. a. Answer: Example 5-3c Another one:

  11. 14-6 Double and Half Angle Formulas

  12. Find the value of if and is between Use the identity First find the value of Subtract. Example 6-1a Example:

  13. Find the square root of each side. Since is in the first quadrant, cosine is positive. Thus, Example 6-1a

  14. Double-angle formula Simplify. Now find Answer: The value of Example 6-1a Example:

  15. Find the value of if and is between Double-angle formula Simplify. Answer: The value of cos 2 Example 6-1a Example:

  16. Find the value of each expression if andis between a.b. Answer: Answer: Example 6-1b Some more:

  17. Find is in the second quadrant. Since we must find first. Example 6-2a Example:

  18. Simplify. Take the square root of each side. Since is in the second quadrant, Half-angle formula Example 6-2a

  19. Simplify the radicand. Rationalize. Multiply. Example 6-2a

  20. Answer: Since is between Thus, is positive and equals Example 6-2a

  21. Find is in the fourth quadrant. Answer: Example 6-2b Yep, there is more:

  22. Find the exact value of by using the half-angle formulas. Example 6-3a Example:

  23. Simplify the radicand. Simplify the denominator. Answer: Example 6-3a

  24. Find the exact value of by using the half-angle formulas. Example 6-3a Example:

  25. Simplify the radicand. Simplify the denominator. Example 6-3a

  26. Answer: Since is in the third quadrant,is negative. Thus, Example 6-3a

  27. Find the exact value of each expression by using the half-angle formulas. a. b. Answer: Answer: Example 6-3b A few more:

  28. Verify that is an identity. Original equation Distributive Property Simplify. Multiply. Example 6-4a Example: Answer:

  29. Verify thatis an identity. Example 6-4b This could be the last one: Answer:

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