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 θ and let r be the distance from the origin to (x, y) θ (x, y) x y r

3. Trig Functions of Acute Angles θ (x, y) x y r

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

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

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

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). 4 5 θ Sin θ = = Cos θ = = Tan θ =

8. Trig Functions of Acute Angles

9. Find the value of the six trig functions of θ whose terminal side passes through (5, 12) 5 12 13 θ Sin θ = Cos θ = Tan θ = Csc θ = Sec θ = Cot θ =

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

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) -4 -3 5 θ Sin θ = Cos θ = Tan θ = Csc θ = Sec θ = Cot θ =

14. Trig Functions of Acute Angles

15. Pythagorean Identities a) Sin 2 θ + Cos 2 θ = 1 b) 1 + Tan 2 θ = Sec 2 θ c) 1 + Cot 2 θ = Csc 2 θ

16. Trig Functions of Acute Angles Sin 2 θ + Cos 2 θ = 1 Csc θ = 3

17. Trig Functions of Acute Angles Csc θ = 3

18. Trig Functions of Acute Angles Sin 2 θ + Cos 2 θ = 1

19. Trig Functions of Acute Angles

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

21. Trig Functions of Acute Angles Cofunction Identities A B C a b c Sin A = = Cos B Tan A = = Cot B Csc A = = Sec 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) Csc θ = 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: a) Soh Cah Toa b) Identities a) Quotient Identities b) Pythagorean Identities c) Cofunction Identities

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

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

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

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

30. Trig Functions of Acute Angles Find the lengths of the missing sides.

31. Trig Functions of Acute Angles 26) Find the length of x. x 45°60° 15 h x'

32. Trig Functions of Acute Angles 26) Find the length of x. x 45°60° 15 x''

33. Trig Functions of Acute Angles 26) Find the length of x. 45°60° 15 X =

34. Trig Functions of Acute Angles 28) Find the length of x. 30° 15° 24 x x' 1

35. Trig Functions of Acute Angles 28) Find the length of x. 30° 15° 24 x 24 -X = 24

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

37. Trig Functions of Acute Angles