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Double-Angle and Half-Angle Identities

Double-Angle and Half-Angle Identities. Section 5.3. Objectives. Apply the half-angle and/or double angle formula to simplify an expression or evaluate an angle. Apply a power reducing formula to simplify an expression. . Double-Angle Identities. Half-Angle Identities.

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Double-Angle and Half-Angle Identities

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  1. Double-Angle and Half-Angle Identities Section 5.3

  2. Objectives • Apply the half-angle and/or double angle formula to simplify an expression or evaluate an angle. • Apply a power reducing formula to simplify an expression.

  3. Double-Angle Identities

  4. Half-Angle Identities

  5. Power-Reducing Identities

  6. Use a half-angle identity to find the exact value of We will use the half-angle formula for sine We need to find out what a is in order to use this formula. continued on next slide

  7. Use a half-angle identity to find the exact value of We now replace a with in the formula to get continued on next slide

  8. Use a half-angle identity to find the exact value of Now all that we have left to do is determine if the answer should be positive or negative. continued on next slide

  9. Use a half-angle identity to find the exact value of We determine which to use based on what quadrant the original angle is in. In our case, we need to know what quadrant π/12 is in. This angle fall in quadrant I. Since the sine values in quadrant I are positive, we keep the positive answer. continued on next slide

  10. If find the values of the following trigonometric functions. For this we will need the double angle formula for cosine In order to use this formula, we will need the cos(t) and sin(t). We can use either the Pythagorean identity or right triangles to find sin(t). continued on next slide

  11. 9 b t 7 If find the values of the following trigonometric functions. Triangle for angle t continued on next slide

  12. 9 t 7 If find the values of the following trigonometric functions. Triangle for angle t Since angle t is in quadrant III, the sine value is negative. continued on next slide

  13. If find the values of the following trigonometric functions. Now we fill in the values for sine and cosine. continued on next slide

  14. If find the values of the following trigonometric functions. For this we will need the double angle formula for cosine We know the values of both sin(t) and cos(t) since we found them for the first part of the problem. continued on next slide

  15. If find the values of the following trigonometric functions. Now we fill in the values for sine and cosine. continued on next slide

  16. At this point, we can ask the question “What quadrant is the angle 2t in?” This question can be answered by looking at the signs of the sin(2t) and cos(2t). is positive and is positive The only quadrant where both the sine value and cosine value of an angle are positive is quadrant I.

  17. If find the values of the following trigonometric functions. We will use the half-angle formula for sine Since we know the value of cos(t), we can just plug that into the formula. continued on next slide

  18. If find the values of the following trigonometric functions. continued on next slide

  19. If find the values of the following trigonometric functions. continued on next slide

  20. If find the values of the following trigonometric functions. Now all that we have left to do is determine if the answer should be positive or negative. In order to do this, we need to know which quadrant the angle t/2 falls in. To do this we will need to use the information we have about the angle t. continued on next slide

  21. If find the values of the following trigonometric functions. The information that we have about the angle t is What we need is information about t/2. If we divide each piece of the inequality, we will get t/2 in the middle of the inequality and bounds for the angle on the left and right sides. continued on next slide

  22. If find the values of the following trigonometric functions. Thus we see that the angle t/2 is in quadrant II. continued on next slide

  23. If find the values of the following trigonometric functions. Since the angle t/2 is in quadrant II, the sine value must be positive. continued on next slide

  24. If find the values of the following trigonometric functions. We will use the half-angle formula for sine Since we know the value of cos(t), we can just plug that into the formula. continued on next slide

  25. If find the values of the following trigonometric functions. continued on next slide

  26. If find the values of the following trigonometric functions. continued on next slide

  27. If find the values of the following trigonometric functions. Now all that we have left to do is determine if the answer should be positive or negative. In order to do this, we need to know which quadrant the angle t/2 falls in. To do this we will need to use the information we have about the angle t. continued on next slide

  28. If find the values of the following trigonometric functions. The information that we have about the angle t is What we need is information about t/2. If we divide each piece of the inequality, we will get t/2 in the middle of the inequality and bounds for the angle on the left and right sides. continued on next slide

  29. If find the values of the following trigonometric functions. Thus we see that the angle t/2 is in quadrant II. continued on next slide

  30. If find the values of the following trigonometric functions. Since the angle t/2 is in quadrant II, the cosine value must be negative.

  31. Use the power-reducing formula to simplify the expression We need to use the power-reducing identity for the cosine and sine functions to do this problem. In our problem the angle a in the formula will be 7x in our problem. We also need to rewrite our problem. continued on next slide

  32. Use the power-reducing formula to simplify the expression Now we just apply the identity to get: continued on next slide

  33. Use the power-reducing formula to simplify the expression This is much simpler than the original expression and the power (exponent) is clearly reduced. continued on next slide

  34. Use the power-reducing formula to simplify the expression Is there another way to simplify this without using a power-reducing formula? The answer to this question is yes. The original expression is the difference of two squares and can be factoring into Now you should notice that the expression in the second set of square brackets is the Pythagorean identity and thus is equal to 1. continued on next slide

  35. Use the power-reducing formula to simplify the expression Is there another way to simplify this without using a power-reducing formula? Now you should notice that what is left is the right side of the double angle identity for cosine where the angle a is 7x. This will allow us to rewrite the expression as

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