1. Engineering math Review Trigonometry Trigonometry Systems of Equations Systems of Equations Vectors Vectors Vector Addition and Subtraction Vector Addition and Subtraction Vector Multiplication Vector Multiplication

2. Trigonometry Review Pythagorean Triples Pythagorean Triples Sine, Cosine, and Tangent revisited. Sine, Cosine, and Tangent revisited. Law of Sines, Law of Cosines Law of Sines, Law of Cosines

3. Starting With Right Triangles a b c

4. a b c Recall the Pythagorean Theorem

5. Starting With Right Triangles a b c

6. Pythagorean Triples There are a few sets of numbers that satisfy the Pythagorean Theorem using only whole numbers for a, b, and c. a b c abc 345 51213 81517

7. Sine a b c

8. Cosine a b c

9. tangent a b c

10. Inverse Trig Functions a b c For example: so then

11. Law of Sines a b c A B C Because of the properties of the sine function between 0 and 180, the Law of Sines may yield 2 answers. If possible, its best to use the Law of Sines when you know whether an angle is acute (less than 90) or obtuse (greater than 90)

12. Law of Cosines a b c A B C The Law of Cosines can be used for obtuse angles (angles greater than 90 degrees) as well as acute angles

13. Law of Cosines a b c A B C The Law of Cosines can be used for obtuse angles (angles greater than 90 degrees) as well as acute angles

14. Law of Cosines a b c A B C The Law of Cosines can be used for obtuse angles (angles greater than 90 degrees) as well as acute angles

15. Example Problem a = 9 b = 18 c = 12 A B C Solve for all angles of the triangle pictured to the left

16. Example Problem a = 9 b = 18 c = 12 A B C Solve for all angles of the triangle pictured to the left Start by using Law of Cosines to determine angle B. This will be the largest angle in the triangle because it is opposite to the longest side.

17. Example Problem a = 9 b = 18 c = 12 A B C Solve for all angles of the triangle pictured to the left Our goal is to calculate angle B. To do this, we’ll need to isolate the expression with angle B in it. We start by squaring both sides of the equation

18. Example Problem a = 9 b = 18 c = 12 A B C Solve for all angles of the triangle pictured to the left Subtract the a 2 and c 2 terms next.

19. Example Problem a = 9 b = 18 c = 12 A B C Solve for all angles of the triangle pictured to the left Divide both sides by -2ac

20. Example Problem a = 9 b = 18 c = 12 A B C Solve for all angles of the triangle pictured to the left Take the inverse cosine of both sides of the equation to isolate the angle of interest, angle B.

21. Example Problem a = 9 b = 18 c = 12 A B C Solve for all angles of the triangle pictured to the left Substitute the known lengths of the sides of the triangle and solve for angle B

22. Example Problem a = 9 b = 18 c = 12 A B C Solve for all angles of the triangle pictured to the left Substitute the known lengths of the sides of the triangle and solve for angle B

23. Example Problem a = 9 b = 18 c = 12 A B C Solve for all angles of the triangle pictured to the left Substitute the known lengths of the sides of the triangle and solve for angle B

24. Example Problem a = 9 b = 18 c = 12 A B = 117.3° C Solve for all angles of the triangle pictured to the left Now that we know one of the angles, and we know that the remaining angles have to be less than 90°, we can use Law of Sines to determine angle A

25. Example Problem a = 9 b = 18 c = 12 A B = 117.3° C Solve for all angles of the triangle pictured to the left Multiplying both sides by ‘a’ isolates the term with angle A in it

26. Example Problem a = 9 b = 18 c = 12 A B = 117.3° C Solve for all angles of the triangle pictured to the left Taking the inverse sine of both sides of the equation solves in terms of angle A

27. Example Problem a = 9 b = 18 c = 12 A B = 117.3° C Solve for all angles of the triangle pictured to the left Substitute known values to solve for angle A

28. Example Problem a = 9 b = 18 c = 12 A B = 117.3° C Solve for all angles of the triangle pictured to the left Substitute known values to solve for angle A

29. Example Problem a = 9 b = 18 c = 12B = 117.3° C Solve for all angles of the triangle pictured to the left We can use Law of Sines to determine angle C or capitalize on the fact that A + B + C = 180°. We’ll use Law of Sines, and then check the results using A + B + C = 180° A = 26.4°

30. Example Problem a = 9 b = 18 c = 12 A = 26.4° B = 117.3° C Solve for all angles of the triangle pictured to the left Multiplying both sides by ‘c’ isolates the term with angle C in it

31. Example Problem a = 9 b = 18 c = 12 A = 26.4° B = 117.3° C Solve for all angles of the triangle pictured to the left Taking the inverse sine of both sides of the equation solves in terms of angle C

32. Example Problem a = 9 b = 18 c = 12 A = 26.4° B = 117.3° C Solve for all angles of the triangle pictured to the left Substitute known values to solve for angle C

33. Example Problem a = 9 b = 18 c = 12 A = 26.4° B = 117.3° C Solve for all angles of the triangle pictured to the left Substitute known values to solve for angle C

34. Example Problem a = 9 b = 18 c = 12B = 117.3° C = 36.3° Solve for all angles of the triangle pictured to the left Now, we can check our answer and see if A + B + C = 180° A = 26.4°

35. Example Problem a = 9 b = 18 c = 12B = 117.3° C = 36.3° Solve for all angles of the triangle pictured to the left Now, we can check our answer and see if A + B + C = 180° A = 26.4°

36. Example Problem a = 9 b = 18 c = 12B = 117.3° C = 36.3° Solve for all angles of the triangle pictured to the left Everything checks out! A = 26.4°