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Math 416

Math 416 . Trigonometry. Time Frame. 1) Pythagoras 2) Triangle Structure 3) Trig Ratios 4) Trig Calculators 5) Trig Calculations 6) Finding the angle 7) Triangle Constructions 8) Word Problems. Right Angle Triangles.

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Math 416

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  1. Math 416 Trigonometry

  2. Time Frame • 1) Pythagoras • 2) Triangle Structure • 3) Trig Ratios • 4) Trig Calculators • 5) Trig Calculations • 6) Finding the angle • 7) Triangle Constructions • 8) Word Problems

  3. Right Angle Triangles • The next section will deal exclusively with right angle triangles. We recall Pythagoras x2 = y2 + z2 Angle Sum θ+ β + = 180° β x y θ z

  4. Pythagoras • Example 72 = x2 + 32 49 = x2 + 9 40 = x2 6.32 = x x = 6.32 7 x 3 Do Stencil #1 

  5. Triangle Structure • We all need to agree on what we are talking about. Consider A <BAC = <A <ABC = <B <BCA = <C AB = c opposite BC = a opposite CA = b opposite b c C B a Do Stencil #2 

  6. Trig Ratios • When we consider the similarity of right angle triangles as long as we ignore decimal angles there are only 45 right angle triangles. • Consider the angles • 90° – 1° – 89° • 90° – 2°– 88° • 90° – 3° – 87° … • 90° – 45° - 45° • Then we start over

  7. Trig Ratios • From ancient times, people have looked at the ratios within right angles triangles • First in tables • Now stored in calculators • We need to define the parts of a right angle triangle • Two types of definitions

  8. Definitions • Absolute – never changes • Relative – involves the position

  9. Absolute vs Relative Now we can define absolutely the hypotenuse as the side opposite the right angle (longest side). In this example it is side AC or b. A Now “relative to angle θ” we define side AB or c as the opposite side Now “relative to angle β” we define side BC or a as the opposite side β b c θ B C a

  10. Absolute Vs. Relative • Now “relative to angle θ” we define side BC or a as the adjacent side • Now “relative to angle β” we define side AB or c as the adjacent side

  11. Labeling the Triangle • Hence with respect to θ Now we define the three main trig ratios… Hyp Opp θ Adj

  12. Trig Ratios • The sine of an angle is defined as the ratio of the opposite to the hypotenuse. Thus Sin θ= Opp Hyp • The cosine of an angle is defined as the ratio of the adjacent to the hypotenuse. Thus Cos θ= Adj Hyp

  13. Trig Ratios • The tangent of an angle is defined as the ratio of the opposite to the adjacent. Thus Tan θ = Opp Adj

  14. SOH – CAH - TOA • You may of heard the acronym SOH – CAH – TOA or SOCK – A – TOA • Sin Opp Hyp • Cos Adj Hyp • Tan Opp Adj

  15. Old Harry And His Old Aunt • There is another acronym… old Harry and his old aunt • Sin Opp Hyp • Cos Adj Hyp • Tan Opp Adj • Use the acronym that you can remember

  16. Example Sin A = 15 39 Sin C = 36 39 • Consider A Cos A = 36 39 Cos C = 15 39 39 36 B C 15 Tan A = 15 36 Tan C = 36 15

  17. Trig Calculator • Now note the table for the assignment is as follows (question #3). For example B 40 24 A C 32 # Angle Sin Cos Tan Angle Sin Cos Tan 32 40 24 40 32 40 24 40 32 24 C 24 32 Eg B

  18. Trig Calculator • We note that these ratios are stored by angle albeit as decimals in a calculator • Note first and foremost your calculators • IT MUST BE IN DEGREES • Make sure you find your DRG (Degree – Radian – Gradients)

  19. Trig Calculator • Hence if θ = 54° then to 4 decimal places • Sin 54° = 0.8090 • Cos 54° = 0.5878 • Tan 54° = 1.3764 Do Stencil #3 

  20. Question #4 • The table required for #4 is as follows • Example θ = 37° • # Sin Cos Tan • Eg 0.6018 0.7986 0.7536

  21. Trig Calculations • There are three basic type of questions. We will focus on the Sine ratio (like question #5) but the techniques are the same for all trig ratio problems.

  22. Trig Calculations Solve for x • Consider Use the angle given to you! 12 Step #1: Determine the Trig Ratio involved with respect to the angle x 40° 12 = hypotenuse, x = opp Thus, SINE

  23. Trig Calculations Step #2 – Determine the equation 12 X = sin 40° 12 x 40° Step #3: Cross multiply (if necessary) x = 12 Sin 40°

  24. Trig Calculations Step #4 If the unknown is isolated (by itself) solve… if not divide then solve 12 x = 7.71 x 40°

  25. Trig Calculations • More Practice x = sin 39° 11 x = 11 sin 39° x = 6.92 11 x 39°

  26. Trig Calculations • Even More Practice 11 = sin 42° x 11 = x sin 42° x 11 42° x = 16.44 Divided both sides by sin 42° or 0.67

  27. Trig Calculations • Even More Practice 9 = sin 73° x x = 9 . sin 73° x = 9.41 x 9 73°

  28. Finding the Angle • Up until now we have the angle get the ratio • Now we need to go the other way • Given the ratio, give the angle • Eg. The buttons we are looking for are the inverse sine (sin -1) • Inverse cosine (cos -1) • Inverse tangent (tan -1) • Find it on your calculator

  29. Examples of Finding the Angle • Find the angle Sin θ = 5 16 θ= sin -1 ( 5 ) 16 θ= 18° (no decimals) 16 5 θ

  30. Another Example • Find the angle sin θ = 7 31 θ= 13° 31 7 θ

  31. Other Examples • Now all the Trig Calculations can follow these procedures x = cos 25° 15 x = 13.59 15 25° x Sin, Cos or Tan?

  32. Another Example • Find the Side x = Tan 61° 6 x = 6 tan 61° x = 10.82 x 61° 6

  33. Another Example • Find the angle sin θ = 7 31 sin θ ( 7 ) 31 θ= 13° 31 7 θ

  34. Another Example • Find the angle cos θ = 5 7 θ = 44° 7 θ 5

  35. Another Example • Find the angle tan θ = 18 5 θ = 74° 18 θ 5

  36. Completing the Triangle • Now using our knowledge we can complete triangles Draw this triangle and another one right below… fill out missing info 76° x 5 = cos 14° x x = 5.15 y = tan 14° 5 y = 1.25 y 14° 5

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