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Section 1.2 Trigonometric Ratios

Section 1.2 Trigonometric Ratios. Objectives: 1. To state and apply the Pythagorean theorem. 2. To define the six trigonometric ratios. A. C. B. Pythagorean Theorem

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Section 1.2 Trigonometric Ratios

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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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