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Trigonometry

Trigonometry. Basic Calculations of Angles and Sides of Right Triangles. Introduction. You can use the three trig functions ( sin , cos , and tan ) to solve problems involving right triangles. 7”. 40°. Introduction. Introduction.

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Trigonometry

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  1. Trigonometry Basic Calculations of Angles and Sides of Right Triangles

  2. Introduction • You can use the three trig functions (sin, cos, andtan) to solve problems involving right triangles.

  3. 7” 40° Introduction Introduction • If you have a right triangle, and you know an acute angle and the length of one side, you have enough info to compute the length of either remaining side. You could compute the length of this side (hypotenuse)... …or this side.

  4. 55 mm 28 mm Introduction Introduction • If you have a right triangle, and you know the lengths of two sides, you have enough info to compute the size of either acute interior angle. …or this angle. You could compute this angle...

  5. Part I Use trigonometry to determine the size of an angle.

  6. Determine an unknown angle • Find the measure of angle A if a=5.5” and b=12”. What is the measure of angle B? • Find the measure of angle A if c=35mm and b=31.5mm • Find the measure of angle A if c=132ft and b=125ft c a A b

  7. Determine an unknown angleExample 1 • To start, we will determine the size of an unknown angle when two sides of the right triangle are known. a=5.5” A b=12”

  8. Determine an unknown angleExample 1 • Let the unknown angle A be the reference angle. 5.5” A 12”

  9. Determine an unknown angleExample 1 • Now label the sides of the right triangle... opposite hypotenuse 5.5” A adjacent 12”

  10. Determine an unknown angleExample 1 • Note that we only know the lengths of the opposite and adjacent sides. opposite hypotenuse 5.5” A adjacent 12”

  11. Determine an unknown angleExample 1 • So we need to pick a trig function that has the opposite and adjacent sides in it... opposite 5.5” A adjacent 12”

  12. Determine an unknown angleExample 1 • Which trig function should you pick? You need to pick the tangent function since it is the only one that has both opposite and adjacent sides in it. 5.5” opposite A 12” adjacent

  13. Now use your calculator to solve. Type-in .458333, press the 2nd function key, then press the tan key Determine an unknown angleExample 1 • Now plug-in the numbers you have into the tangent function... A = 24.6° opposite 5.5” A adjacent 12”

  14. Determine an unknown angleExample 1 • How could you determine the size of the remaining angle? …this one must be 65.4° degrees. (Since 180° - 90° - 24.6° = 65.4°) 65.4° ..and this one was computed to be 24.6°… 5.5” This angle is 90°… 24.6° 12”

  15. 35 mm A 31.5 mm Determine an unknown angleExample 2 • Let’s try another one… • Determine the size of angle A.

  16. Determine an unknown angleExample 2 • First, label the sides of the triangle... hypotenuse 35 mm opposite A 31.5 mm adjacent

  17. Determine an unknown angleExample 2 • Since you know the lengths of the adjacent side and the hypotenuse, pick a trig function that has both of these... hypotenuse 35 mm A 31.5 mm adjacent

  18. Determine an unknown angleExample 2 • Which trig function should you pick? You need to pick the cosine function since it is the only one that has both the adjacent side and hypotenuse in it. hypotenuse 35 mm A 31.5 mm adjacent

  19. Now use your calculator to solve. Type-in 0.9, press the 2nd function key, then press the cos key Determine an unknown angleExample 2 • Now plug-in the numbers you have into the cosine function... hypotenuse 35 mm A 31.5 mm adjacent

  20. Determine an unknown angleExample 2 • Now that you know how big angle A is, determine the size of the remaining angle. 35 mm 25.8° 31.5 mm

  21. Determine an unknown angleExample 2 • To determine the other angle: • 180° - 90° - 25.8° = 64.2° 64.2° 35 mm 25.8° 31.5 mm

  22. 125 mm 132 mm A Determine an unknown angleExample 3 • Let’s try one more. • Determine the size of angle A.

  23. 125 mm 132 mm A Determine an unknown angleExample 3 • Label the sides of the triangle... opposite adjacent hypotenuse

  24. 125 mm 132 mm A Determine an unknown angleExample 3 • Since you know the lengths of the opposite side and the hypotenuse, pick a trig function that contains them... opposite hypotenuse

  25. Determine an unknown angleExample 3 • Which trig function should you pick? You need to pick the sine function since it is the only one that has both the opposite side and hypotenuse in it. opposite 125 mm 132 mm hypotenuse A

  26. 125 mm Now use your calculator to solve. Type-in 0.947, press the 2nd function key, then press the sin key 132 mm A Determine an unknown angleExample 3 • Now plug-in the numbers you have into the sine function... opposite hypotenuse

  27. Determine an unknown angleExample 3 • What is the size of the remaining angle? 125 mm 132 mm 71.3°

  28. Determine an unknown angleExample 3 • The angle is computed to be 18.7°. 125 mm 18.7° 132 mm 71.3°

  29. Summary of Part I • By now you should feel like you have a pretty good chance of determining the size of an angle when any two sides of a right triangle are known. • Click to see one more problem like the last three you have done...

  30. Summary of Part IExample 4 • Determine the size of angle A. • Solve the problem, then click to see the answer. 25.5 ft A 23 ft

  31. Summary of Part IExample 4 • Selecting the cos function will allow you to determine the size of angle A. hypotenuse 25.5 ft A 23 ft adjacent

  32. Part II Use trigonometry to determine the length of a side of a right triangle.

  33. 7” 40° Determining the length of a side • Recall that if you have a right triangle, and you know an acute angle and the length of one side, you have enough info to compute the length of either remaining side. You could compute the length of this side (hypotenuse)... …or this side.

  34. Determine an unknown side • Find the measure of side a if angle A =260 and c=9”? • Find the measure of side b if angle A =470 and c=75mm? • Find the measure of side c if angle A =350 and a=12”? c a A b

  35. Determining the length of a sideExample 5 • In this problem, we will determine the length of side x. 9” x 26°

  36. Determining the length of a sideExample 5 • As always, first label the sides of the triangle... hypotenuse 9” opposite x 26° adjacent

  37. Determining the length of a sideExample 5 • Since you know the length of the hypotenuse and want to know the length of the opposite side, you should pick a trig function that contains both of them... hypotenuse 9” opposite x 26°

  38. Determining the length of a sideExample 5 • Which trig function should you pick? You need to pick the sine function since it is the only one that has both the opposite side and hypotenuse in it. hypotenuse 9” x opposite 26°

  39. Use basic algebra to solve this equation. Multiply both sides of the equation by 9 to clear the fraction. Determining the length of a sideExample 5 • Now set-up the trig function: hypotenuse 9” opposite x 26°

  40. Determining the length of a sideExample 5 • Now you know the opposite side has a length of 3.95”. hypotenuse 9” opposite 3.95” 26°

  41. x 75 mm 47° Determining the length of a sideExample 6 • Let’s try another one. • Determine the length of side x.

  42. x 75 mm 47° Determining the length of a sideExample 6 • Since the known angle (47°) will serve as your reference angle, you can label the sides of the triangle... opposite adjacent hypotenuse

  43. x 75 mm 47° Determining the length of a sideExample 6 • You know the length of the hypotenuse and want to know the length of the adjacent side, so pick a trig function which contains both of them... adjacent hypotenuse

  44. x 75 mm 47° Determining the length of a sideExample 6 • Which trig function should you pick? You need to pick the cosine function since it is the only one that has both the adjacent side and hypotenuse in it. adjacent hypotenuse

  45. Use basic algebra to solve this equation. Multiply both sides of the equation by 75 to clear the fraction. To finish, evaluate cos 47° (which is 0.682) and multiply by 75. x 75 mm 47° Determining the length of a sideExample 6 • Set-up your trig function... adjacent hypotenuse

  46. Determining the length of a sideExample 6 • Now you know the length of the adjacent side is 51.1 mm. 51.1 mm 75 mm adjacent hypotenuse 47°

  47. Determining the length of a sideExample 7 • Let’s try a little bit more challenging problem. • Determine the length of side x. x 12 ft 35°

  48. Determining the length of a sideExample 7 • Label the sides of the right triangle... hypotenuse x opposite 12 ft 35° adjacent

  49. Determining the length of a sideExample 7 • Which trig function will you pick? You know the length of the side opposite and want to know the length of the hypotenuse. hypotenuse x opposite 12 ft 35° adjacent

  50. Determining the length of a sideExample 7 • Which trig function should you pick? You need to pick the sine function since it is the only one that has both the opposite side and hypotenuse in it. hypotenuse opposite x 12 ft 35°

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