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4 -6 The Law of Sines

4 -6 The Law of Sines. Chapter 4 The Trigonometric Functions. Right triangle trigonometry can be used to solve problems involving right triangles. However, many interesting problems involve non-right triangles.

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4 -6 The Law of Sines

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  1. 4-6 The Law of Sines Chapter 4 The Trigonometric Functions

  2. Right triangle trigonometry can be used to solve problems involving right triangles. However, many interesting problems involve non-right triangles. Today, you will use right triangle trigonometry to develop the Law of Sines.

  3. B c a A b C

  4. Objectives • Solve triangles by using the Law of Sines if the measures of two angles and a side is given, or two sides and an angle. (not SSA) • Find the area of a triangle if the measures of two sides and the included angle or the measures of two angles and a side are given.

  5. The Law of Sines… …can be used to solve triangles that are not right triangles. The Law of Sines for any triangle ABC…

  6. Example 1 Solve LMN if L = 29, M = 112. And l = 22

  7. Example 1 You try… Solve ABC if A = 40, B = 59. And c = 14

  8. Example 2 A person in a hot-air balloon observes that the angle of depression to a building on the ground is 65.8. After ascending 500 feet, the person now observes that the angle of depression is 70.2. How far is the balloonist now from the building? (Sketch a picture and use the Law of Sines to solve)

  9. Finding Areas of Triangles SAS Two ways First: If two sides and an included angle are known, use one of the following:

  10. Example 3 • Find the area of ABC if b = 21.2, c = 16.5, and A = 25 (Since we know 2 sides and an angle, we will use one of the first three formulas.) Hint: Label diagrams carefully!

  11. Example 3 You try… • Find the area of ABC if a = 68, c = 110, and B = 42.5 (Since we know 2 sides and an angle, we will use one of the first three formulas.) Hint: Label diagrams carefully!

  12. Finding Areas of Triangles ASA Two ways Second: If two angles and an one side are known, use one of the following:

  13. Example 4 • Find the area of JKL if j = 45.7, K = 111.1, and L = 27.3. (Since we know two angles and one side, we will use one of the second three formulas) Hint: Label diagrams carefully!

  14. Example 4 You try… • Find the area of ABC if c = 4, A = 37, and B = 69. (Since we know two angles and one side, we will use one of the second three formulas) Hint: Label diagrams carefully!

  15. Assignment • P. 298, 1, 4, 5, 7, 45, 46 (only 6problems)

  16. What is deliberate practice? • 1. Deliberate practice is highly demanding mentally, requiring high levels of focus and concentration.You’ve heard it before – no pain, no gain. Research indicates that you have to be “fully absorbed” in your practice for it to truly be effective. This is suggestive of the concept of “flow.”

  17. What is deliberate practice? • 2. It is designed specifically to improve performance—to strengthen it beyond its current levels.This is the part that says you can’t just put in time and expect to get significantly better at anything – you have to consistently stretch yourself, and then stretch some more.

  18. What is deliberate practice? • 3. It must continue for long of periods of time.This is Gladwell’s 10,000 hours/10 years. The authors go on to say “Basic research on expert performance suggests that the benefits it generates cannot usually be attained with less than 10 years of continued, vigorous effort (e.g., Ericsson, 2006).”

  19. What is deliberate practice? • 4. It must be repeated.Even though repetition alone won’t get you to the level of excellence, you also won’t get there without out it. Perhaps this why the word “Practice” is repeated three times in the old joke.

  20. What is deliberate practice? • 5. It requires continuous feedback on results.Sometimes you can tell on your own whether you are doing things right. I know when I hit a wrong note on the guitar, for example. But very often this is the area where having a great teacher, coach, or mentor can make all the difference.

  21. What is deliberate practice? • 6. Pre-performance preparation is essential.This is where goal setting comes in – you have to know where you want to go if you expect to get there. And as the authors stress, goal-setting “should involve not merely outcomes, but also the processes involved in reaching predetermined goals.”

  22. What is deliberate practice? • 7. It involves self-observation and self-reflection.As you practice, you need to be continually aware of your own performance and be focused on correcting and adapting as appropriate.  This kind of in-the-moment self-assessment is critical regardless of whether a teacher is involved.

  23. What is deliberate practice? • 8. It involves careful reflection on performance after practice sessions are completed.In addition to being aware of your performance as you are practicing, you need to look back on it once you are done and determine where you stand with respect to your overall goals. What might you change the next time to ensure ongoing progress? From Mission to Learn, by Jeff Cobb http://www.missiontolearn.com/2010/04/deliberate-practice/

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