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Understanding Trigonometric Ratios: Pythagorean Theorem and Six Basic Functions

This educational resource covers essential concepts in trigonometry, including the Pythagorean theorem and the six trigonometric ratios: sine, cosine, tangent, cosecant, secant, and cotangent. The Pythagorean theorem is illustrated in the context of right triangles, highlighting the relationship between the lengths of the sides. The six trigonometric functions are defined using the sides of right triangles, and examples demonstrate how to calculate these ratios for various angles. Practice questions are included to reinforce understanding.

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Understanding Trigonometric Ratios: Pythagorean Theorem and Six Basic Functions

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  1. Section 1.2 Trigonometric Ratios

  2. Objectives: 1. To state and apply the Pythagorean theorem. 2. To define the six trigonometric ratios.

  3. A C B Pythagorean Theorem In right ABC, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. a2 + b2 = c2

  4. leg opposite A sine of A = hypotenuse leg adjacent A cosine of A = hypotenuse leg opposite A tangent of A = leg adjacent A Trigonometric Ratios

  5. SOHCAHTOA ine pposite ypotenuse osine djacent ypotenuse angent pposite djacent

  6. o a o sin A cos A tan A = = = h h a Trigonometric Ratios hypotenuse opposite adjacent A

  7. 1 cosecant of A = cscA = sinA 1 secant of A = secA = cosA 1 cotangent of A = cotA = tanA Reciprocal Ratios

  8. hypotenuse opposite adjacent A h h a csc A sec A cot A = = = o a o Reciprocal Ratios

  9. g2 + e2 = f2 g2 + 62 = 82 g2 + 36 = 64 g2 = 28 g = 2 7 EXAMPLE 1 Find the six trigonometric ratios for G in right EFG. G 8 6 E F

  10. G 11 9 2 10 11 9 10 20 2 10 9 9 11 E F Practice Question: Find the six trigonometric ratios for E in right EFG. sin E = 1. 2. 3. 4.

  11. P(x,y) r y  x

  12. opp y hyp r sin q = = csc q = = hyp r opp y adj x hyp r cos q = = sec q = = hyp r adj x opp y adj x tan q = = cot q = = adj x opp y Trigonometric Ratios

  13. EXAMPLE 2 Find the six trigonometric ratios for a 90º angle. P = (0, 1) x=0, y=1, r=1 cos  = 0 sin  = 1 tan  = und. 90° P 

  14. EXAMPLE 2 Find the six trigonometric ratios for a 90º angle. P = (0, 1) x=0, y=1, r=1 sec  = und. csc  = 1 cot  = 0 90° P 

  15. Practice Question: Find the six trigonometric ratios for a 180º angle. sin  = 1. -1 2. 0 3. 1 4. und. 180°  P

  16. 45 c = 2 1 45 1 Special Triangles a2 + b2 = c2 12 + 12 = c2 1 + 1 = c2 c2 = 2

  17. 1 æ ö 2 ç ÷ 2 ç ÷ 2 è ø 45 c = 2 1 45 1 Special Triangles sin 45° =

  18. 2 2 2 2 45 c = 2 1 45 1 Special Triangles sin 45° = cos 45° = tan 45° = 1

  19. 2 2 45 c = 2 1 45 1 Special Triangles csc 45° = sec 45° = cot 45° = 1

  20. 2 2 2 2 2 sin 45° = csc 45° = 2 cos 45° = sec 45° = tan 45° = 1 cot 45° = 1

  21. 30 2 b = 3 60 1 Special Triangles a2 + b2 = c2 12 + b2 = 22 1 + b2 = 4 b2 = 3

  22. 30 æ ö 1 3 2 ç ÷ ç ÷ b = 3 3 3 è ø 60 1 Special Triangles tan 30° =

  23. 30 1 2 b = 3 2 3 3 60 2 3 1 Special Triangles tan 30° = sin 30° = cos 30° =

  24. 3 30 2 b = 3 2 3 60 3 1 Special Triangles cot 30° = csc 30° = 2 sec 30° =

  25. 3 tan 30° = cot 30° = 1 2 sin 30° = csc 30° = 2 3 3 2 3 cos 30° = sec 30° = 2 3 3

  26. 30 1 2 b = 3 2 3 60 2 1 3 Special Triangles sin 60° = cos 60° = tan 60° =

  27. 30 2 b = 3 3 2 3 60 3 3 1 Special Triangles csc 60° = sec 60° = 2 cot 60° =

  28. sin 60° = csc 60° = 1 2 cos 60° = sec 60° = 2 3 3 2 3 tan 60° = cot 60° = 2 3 3 3

  29. Homework: pp. 12-13

  30. 21 ►A. Exercises 1. Find the six trig. ratios for both acute angles in each triangle. A 5 2 B C

  31. = = n 140 2 35 ►A. Exercises 3. Find the six trig. ratios for both acute angles in each triangle. n M L 2 12 N 1st-solve for side n 22 + n2 = 122 n2 = 122-22 = 144-4 = 140

  32. 2 35 #3. M L 2 12 N

  33. 37 ►A. Exercises 7. Find the six trig. functions for an angle in standard pos. whose terminal ray passes through the point (-6, -1). -6  -1

  34. -6  -1 37 sin  = csc  = cos  = sec  = tan  = cot  =

  35. ►B. Exercises 15. Find the six trig. ratios for the quadrantal angle measuring 180°.

  36. ►B. Exercises 15. Find the six trig. ratios for the quadrantal angle measuring 180°. (-1, 0)

  37. ■ Cumulative Review 30. Give the distance between (2, 7) and (-3, -1).

  38. ■ Cumulative Review 31. Give the midpoint of the segment joining (2, 7) and (-3, -1).

  39. ■ Cumulative Review 32. Give the angle  coterminal with 835 if 0    360.

  40. ■ Cumulative Review 33. Convert 88 to radians.

  41. ■ Cumulative Review 34. If sec  = 7, find cos .

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