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Trig Functions of acute angles

Trig Functions of acute angles. Section 12-2 Pages 555-560. Trig Functions of Acute Angles. Draw an acute angle θ in standard position. Choose any point (x, y) on the terminal side of θ and let r be the distance from the origin to (x, y). (x, y). r. y. θ. x.

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Trig Functions of acute angles

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  1. Trig Functions of acute angles Section 12-2 Pages 555-560

  2. Trig Functions of Acute Angles • Draw an acute angle θ in standard position. • Choose any point (x, y) on the terminal side of θand let r be the distance from the origin to (x, y) (x, y) r y θ x

  3. Trig Functions of Acute Angles • With this triangle, we have the following definitions: Sin θ = Cos θ = Tan θ = (x, y) r y θ x

  4. Trig Functions of Acute Angles • Another way to think of this is using the word: Cah Toa Soh Sin θ = Hypotenuse Opposite θ Adjacent

  5. Trig Functions of Acute Angles • Another way to think of this is using the word: Cah Toa Soh Cos θ = Hypotenuse Opposite θ Adjacent

  6. Trig Functions of Acute Angles • Another way to think of this is using the word: Cah Toa Soh Tan θ = Hypotenuse Opposite θ Adjacent

  7. Trig Functions of Acute Angles • Find the Sin θ, Cos θ, and Tan θ given that θ is an angle in standard position passing through the point (4, 5). Sin θ = = 5 θ Cos θ = = 4 Tan θ =

  8. Trig Functions of Acute Angles • In addition to the Sin θ, Cos θ, and Tan θ, there are three reciprocal functions • Cosecant θ(Cscθ) = • Secant θ (Sec θ) = • Cotangent θ (Cot θ) =

  9. Trig Functions of Acute Angles • Find the value of the six trig functions of θ whose terminal side passes through (5, 12) Sin θ = Cscθ = 13 12 Cos θ = Sec θ = θ Tan θ = Cot θ = 5

  10. Trig Functions of Acute Angles • Find the value of the six trig functions of θ whose terminal side passes through (2, 4) Sin θ = Cscθ = Cos θ = Sec θ = 2 4 θ 2 Tan θ = Cot θ = 2

  11. Trig Functions of Acute Angles • Oral Exercises 1-4 • Page 559, 1-8

  12. Trig Functions of acute angles Section 12-2 Pages 555-560

  13. Trig Functions of Acute Angles • Find the value of the six trig functions of θ whose terminal side passes through (-3,- 4) -3 Sin θ = Cscθ = θ -4 5 Cos θ = Sec θ = Tan θ = Cot θ =

  14. Trig Functions of Acute Angles • A trigonometric equation that is true for all values of θ is called a trigonometric identity • Quotient Identities Tan θ = Cot θ =

  15. Trig Functions of Acute Angles • Pythagorean Identities • Sin2θ + Cos2θ = 1 • 1 + Tan2θ = Sec2θ • 1 + Cot2θ = Csc2θ

  16. Trig Functions of Acute Angles • Using the identities, find the value of the remaining trig functions if Sin θ = Sin2θ + Cos2θ = 1 Cscθ = 3 2+ Cos2θ = 1 = Sec θ = + Cos2θ = 1 Cos2θ = Cosθ=

  17. Trig Functions of Acute Angles • Using the identities, find the value of the remaining trig functions if Sin θ = Sinθ= Cscθ = 3 Cosθ= Sec θ = Tanθ= Cot θ = 2

  18. Trig Functions of Acute Angles • Using the identities, find the value of the remaining trig functions if Cosθ = Sin2θ + Cos2θ = 1 Cscθ = 2θ+ 2= 1 Sec θ = Sin2θ + = 1 Sin2θ = Sinθ=

  19. Trig Functions of Acute Angles • Using the identities, find the value of the remaining trig functions if Sin θ = Sinθ= Cscθ = Cosθ= Sec θ = Tanθ= Cot θ=

  20. Trig Functions of Acute Angles • Cofunction Identities • The Sine and Cosine functions are called cofunctions. B What can we say about angles A and B? → Complementary c a Sin A = = Cos B C A b Cos B =

  21. Trig Functions of Acute Angles • Cofunction Identities Sin A = = Cos B B Tan A = = Cot B c a Csc A = = Sec B C A b

  22. Trig Functions of Acute Angles • Cofunction Identities From these identities, note that: Sin θ = Cos (90° – θ) Tan θ = Cot (90° – θ) Sec θ = Csc(90° – θ) Cos θ = Sin (90° – θ) Cot θ = Tan (90° – θ) Cscθ = Sec (90° – θ)

  23. Trig Functions of Acute Angles • Oral Exercises 5-8 • Page 559, 9-19

  24. Trig Functions of acute angles Section 12-2 Pages 555-560

  25. Trig Functions of Acute Angles • So far in this section: • SohCah Toa • Identities • Quotient Identities • Pythagorean Identities • Cofunction Identities

  26. Trig Functions of Acute Angles • In trigonometry, there are two triangles that are used repeatedly in different problems. • 30-60-90 right triangle • 45-45-90 right triangle 60° 2 1 30° 45° 1 45° 1

  27. Trig Functions of Acute Angles • Using these two triangles, we can find the following: 30° 45° 60° Sin θ Cos θ Tan θ

  28. Trig Functions of Acute Angles • Use the two triangles (or chart) to evaluate the reciprocal functions. 30° 45° 60° Cscθ 2 2 Sec θ Cot θ

  29. Trig Functions of Acute Angles • Find the lengths of the missing sides. Cos 24

  30. Trig Functions of Acute Angles • Find the lengths of the missing sides. Sin 5 10

  31. Trig Functions of Acute Angles • Find the length of x. Sin 15 2 15 h 45° 60° x x' x'

  32. Trig Functions of Acute Angles • Find the length of x. Tan 15 45° 60° x x''

  33. Trig Functions of Acute Angles • Find the length of x. X = 15 = 45° 60°

  34. Trig Functions of Acute Angles • Find the length of x. 1 Tan x 24 x' 15° 30° 24

  35. Trig Functions of Acute Angles • Find the length of x. Tan x 24 8 24 15° x 8 X = 24 - 30° 24

  36. Trig Functions of Acute Angles • Oral Exercises 9-12 • Page 559, 20-34

  37. Trig Functions of Acute Angles

  38. Trig Functions of Acute Angles

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