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TRIGONOMETRY

TRIGONOMETRY. Understand that the similarity of triangles is fundamental to the trigonometric functions, sin theta, cos theta and tan theta, and is able to define and use the functions (LO 3 AS 5) Solve problems in two dimensions by using the trig

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TRIGONOMETRY

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  1. TRIGONOMETRY

  2. Understand that the similarity of triangles is fundamental to the trigonometric functions, sin theta, cos theta and tan theta, and is able to define and use the functions (LO 3 AS 5) Solve problems in two dimensions by using the trig functions in right – angled triangles and by constructing and interpreting geometric and trigonometric models. (LO 3 AS 6) Demonstrate the ability to work with various types of functions. (LO 2 AS 1 a)

  3. Recognize relationships between variables in terms of numerical, graphical, verbal and symbolic representations. (LO 2 AS 1b) Generate as many graphs as possible, initially by means of point – point plotting, supported by available technology, to make and test conjectures and hence to generalize the effects of the parameters a and q on the graphs of the functions including:

  4. Similar Triangles • Before discussing the concepts of trigonometry in this module, we will need to briefly revise the concept of similar triangles. • If two triangles are similar, their corresponding angles are equal and their corresponding sides are in proportion.

  5. If then: (a) (b) Example: In the following triangles, show that if , then

  6. Introduction to Trigonometry • Trigonometry is the study of the relationships between lines and triangles. It has its origins in the study of astronomy where distances cannot be measured directly but have to be calculated. Evidence of such calculations has been found on ancient Babylonian clay tablets and in the relics of the ancient Egyptian civilization. Angles in Trigonometry are usually indicated by means of Greek letters: = theta, = beta, = alpha

  7. Right-angled triangles are fundamental to the study of trigonometry.

  8. The side AC, which lies opposite the right angles, is called the hypotenuse. The side AB lies opposite and side BC lies adjacent (next to) the angle .

  9. Investigation 1 In the diagram which follows on the next page, (a) For each similar triangle, measure the length of the side opposite the 30 angle, adjacent to the 30° angle and the hypotenuse. Record your results in the table below. (b) Calculate the ratios in the table using a calculator. Round off to one decimal place if necessary.

  10. What can you conclude?

  11. Investigation 2 In the diagram which follows on the next slide (a) For each similar triangle, measure the length of the side opposite the angle, adjacent to the 30°angle and the hypotenuse. Record your results in the table below. (b) Calculate the ratios in the table using a calculator. Round off to one decimal place if necessary.

  12. Summary of conclusions from the two investigations: • For any constant angle, the ratios for each similar triangle remain the same. • As the length of the hypotenuse increases, the length of the adjacent and opposite sides increase in the same proportion.

  13. Definitions of the trigonometric ratios

  14. The ratio is called the sine of the angle. • This can be written as sin = • The ratio is called the cosine of the angle • This can be written as cos = • The ratio is called the tangent of the angle . • This can be written as tan = • Remember: A trigonometric ratio is a numerical value and not an angle

  15. Example Consider the following diagram and then answer the questions that follow.

  16. Complete: (a) sin = (b) cos = (c) tan = (d) sin (e) cos (f) tan

  17. EXERCISE 1 • Write down the following: (a) sin C (b) cos C (c) tan C (d) sin B (e) cos B (f) tan B

  18. 2. Write down the following: • sin • cos • tan • sin • cos (f) tan

  19. Evaluating the value of trigonometric ratios • If you ensure that your calculator is on the DEG mode, you can evaluate trigonometric ratios without having to draw triangles as done in the previous investigations. • Consider the first investigation done in this topic. It is clear from that investigation that sin .

  20. We can use a calculator to do this calculation for us. • On your calculator, press the button “sin” and then “30” and then “= “. Guess what? • You get 0, 5. Awesome, isn’t it! • This means that we can work out trigonometric ratios for any angle.

  21. Examples Use a calculator to evaluate the following trigonometric ratios rounded off to two decimal places where necessary: (a) cos 20° = (b) tan 10° = (c) sin 30° = (d) sin 47° = (e) cos l46° = (f) tan 235° = (g) 3 cos 20° = (h) • (j) • (l) What do you notice about (k) and (l)?

  22. EXERCISE 2 1. Evaluate the following rounded off to two decimal places where appropriate: (a) sin 57° (b) tan 67° (c) cos l24° (d) cos 320° (e) 3 sin 45° (f) 7tan 58° (g) sin 130° (h) 13 tan (45° + 54°) (i) 25 sin 225°

  23. (j) (k) (l) (m)

  24. 2. If means the same as , use this idea to calculate the value of the following rounded off to three decimal places: (a) sin 309° (b) sin 56° + cos 56° (c) sin 162° + cos 162° (d) sin 46° + cos 65° (e) Cos 32° (f) sin 124° (g) tan 124° (h) tan l35° (i)5cos 25° What conclusion can you make from the above exercise?

  25. Calculating the size of an angle when given the trigonometric ratio • Consider the equation sin = 0, 5. • Here we want to find the angle that gives the number 0, 5. • In order to do this, we will make use of the button sin on the calculator. • If sin = 0, 5, then we can find by using the sequence:

  26. Some calculators use the button INV or SHIFT instead of 2nd F. • You need to make sure that you know how to use your calculator to do this work.

  27. Examples • Solve the following equations: (Round your answers off to two decimal places when necessary.) (a) cos = 0, 5 (b) 2sin = 1,124 (c) tan - 4,123 = 0 (d) cos 2 = 0,435

  28. (e) (f)

  29. EXERCISE 3 • Solve the following equations: (Round your answers off to two decimal places when necessary). (a) (b) (c) (d) (e)

  30. (f) (g) (h) (i) (j) (k) (l)

  31. Solving problems using trigonometric ratios • Type 1 • (Calculating the length of a side when given an angle and another side) • Example 1 Calculate the length of AB in • You want side AB, which is opposite 36°. • You have side BC, the hypotenuse.

  32. You now need to create an equation involving the ratio and the angle 36°: sin 36°

  33. Example 2 Calculate the length of BC to one decimal place. • You want side BC, which is adjacent to 59°. • You have side AB, which is opposite to 59°. You now need to create an equation involving the ratio and the angle 59°:

  34. = tan 59degrees

  35. EXERCISE 4 (Round answers off to one decimal place in this exercise) • Calculate the length of PQ in

  36. 2. (a) Calculate the length of AB. (b) Calculate the length of BC. (c) What is the size of ?

  37. 3. By using the information provided on the diagram, calculate: (a) the length of AC. (b) the length of AB.

  38. 4. In the diagram, BD AC. Using the information provided, calculate the length of AC.

  39. 5. Using the information provided on the diagram , calculate the length of BC.

  40. Type 2 Calculating the size of an angle when given two sides • Example 4 • Calculate the size of to one decimal place. • We need to find angle . • We have side BC, which is adjacent to Side AC is the hypotenuse.

  41. Therefore, we need to form an equation involving the ratio and the angle .

  42. EXERCISE 5 • Round answers off to one decimal place in this exercise • (a) Calculate the size of . (b) Calculate the length of AC.

  43. 2. (a) Calculate the size of . (b) Calculate the size of .

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