1. Trigonometry

2. Definition:
Trigonometry is the art of studying triangles (in particular, but not limited to, right triangles)
Trigonometry makes use of both the angles and the side lengths
Deals with the relationships between the angles and side lengths of a triangle

3. The sine, cosine and tangent of acute angles in a 90° triangle show how the side lengths are related to the angles

4. Before learning the key formulas in trigonometry, it is of absolute importance that some terms are understood
Because we are dealing with right triangles, you are already familiar with one very important right triangle theorem:
The Pythagorean Theorem
a² + b² = c²

5. In every right triangle, because one of the angles measures 90°, then logically the other two angles must add up to 90°
Because m<B = 90° then m<A + m<C = 90° (since there are 180° in every triangle)

6. Hypotenuse: Opposite Side: Adjacent Side:
The side that is opposite the right angle
The longest side in the right triangle
Opposite Side:
The side that is opposite of a given angle
Ex: Side AB is opposite m<C
Side BC is opposite m<A
Adjacent Side:
The side that is neither hypoteneuse or opposite
Ex: Side BC is adjacent to m<C
Side AB is adjacent to m<A

7. Example: Fill in the blanks in the following questions:
Hypotenuse: _________________
Opposite m<A: _________________
Adjacent m<A: _________________
Opposite m<C: __________________

8. These three definitions of the sides are of utmost importance in trigonometry
They are at the root of finding every angle in a right triangle
Class work and homework:
Hand out 1 and Math 3000 page 182 # 1

9. Trigonometric Ratios in a Right Triangle

10. Sine of an acute angle
The sine of an acute angle is equal to the ratio of the measure of the opposite side to that angle over the measure of the hypotenuse
The sine of angle A is written sin A

11. Cosine of an Acute angle
The cosine of an acute angle is equal to the ratio of the measure of the adjacent side to that angle over the measure of the hypotenuse
The cosine of angle A is written cos A

12. Tangent of an acute angle
The tangent of an acute angle is equal to the ratio of the measure of the opposite side to that angle over the measure of the adjacent side to that angle
The tangent of angle A is written tan A

13. The MOST important Gibberish word you will need to remember in life
SOH – CAH - TOA

15. Example:
60° 2 1
30°

16. When writing a trigonometric ratio, we can write the measure of the angle when it is known.
Thus:
The sine of angle B measuring 30° is written sin 30°

17. Homework
P. 182 # 1
P. 184 # 2

18. (2) Review Definitions: Trig Ratios : SOH CAH TOA
Hand out 1 – Identifying Opposite, Adjacent, and Hypotenuse & Sine, Cosine, Tangent Problems
Hand out 2 – Find the Trig Ratio

19. 3. Using your Calculator

20. Using Your Calculator
The keys sin, cos, tan on the calculator enable you to calculate the value of sin A, cos A, or tan A knowing the measure of angle A
So if you know the measure of an angle you can use the sin, cos, or tan buttons on your calculator in order to calculate its value

21. Formulas for 90° triangle
Formulas to find a missing side
Formulas to find a missing angle
(hyp)² = (opp)² + (adj)²

22. Summary
The three new formulas for 90° triangles Sin, Cos, Tan are used to find a missing side length in a right triangle

23. 2. The keys sin-1, cos-1, tan-1 on the calculator enable you to calculate the measure of angle A knowing sin A
So if know sin A, cos A, or tan A, you can calculate the measure of angle A

24. Summary
If we take the inverse of each formula, we can find the missing side angle in a 90° triangle
The symbol for the inverse of
sin (A) is sin-1; cos (A) is cos-1;
tan (A) is tan-1

25. Example
sin 30º = 0.5 and sin-1 (0.5) = 30º

26. Class work Mathematics 3000 , page 185 numbers 4 and 5
Hand out number 2

27. Finding Missing Angles using Trigonometry Ratios

28. Formulas for 90° triangle
Formulas to find a missing side
Formulas to find a missing angle
(hyp)² = (opp)² + (adj)²

29. In a Right Triangle
Find the acute angle A when its opposite side and the hypotenuse are known requires the use of sin A
SOH – Opposite/hypotenuse
sin A = M<A=sin-1 ( )=53.1º

30. Finding the acute angle A when its adjacent side and the hypotenuse are known requires the use of cos A
Cos = adjacent/hypotenuse
Cos A= m<A = cos-1 ( ) = 41.4º

31. Finding the acute angle A when its opposite side and adjacent side are known requires the use of tan A
tan = opposite/adjacent
Tan A = m<A=tan-1 ( ) = 56.3º

32. How to: Label known sides H,O,A
Select sine, cosine or tangent depending on information known
Set up ratio – leave either as a ratio OR reduce to decimal round to 4 places (thousandth)
M<a = inverse of sine, cos or tan and that ratio or decimal
Result is your missing angle

33. Class work Mathematics 3000 page 186, activity 4
Mathematics 3000, page 187, number 8
Handout number 4

34. Finding Missing Sides Using
Trigonometric Ratios

35. Formulas for 90° triangle
Formulas to find a missing side
Formulas to find a missing angle
(hyp)² = (opp)² + (adj)²

36. In a right triangle Sin 50º= x=5sin50º = 3.83 cm
Finding the measure x of side BC opposite to the known angle A, knowing also the measure of the hypotenuse, requires the use of sin A
Remember: SOH
*****Cross Multiply*****
Sin 50º= x=5sin50º = 3.83 cm

37. Finding the measure y of side AC adjacent to the known Angle A, knowing also the measure of the hypotenuse, requires the use of cos A Remember: cos = adjacent/hypotenuse *****Cross Multiply***** Cos 50º = y = 5 cos 50º = 3.21 cm

38. Finding the measure x of side BC opposite to the known angle A, knowing also the measure of the adjacent side to angle A, requires the use of tan A
remember tan=opposite/adjacent
***cross multiply***
tan 30º = x = 4 tan 30º = 2.31 cm

39. Class work Mathematics page 185, activity 3
Mathematics page 186, numbers 6,7
Handout number 6

40. Class Work and Homework
Page 186, numbers 6 and 7

41. To determine the measure of all its sides and angles
Solving a triangle
To determine the measure of all its sides and angles

42. Class work and homework
Math3000 page187, number 9
Page 188, number 10, 11,12

43. Sine Law

44. The sides in a triangle are directly proportional to the sine of the opposite angles to these sides

45. It is also true that:

46. The sine law can be used to find the measure of a missing side or angle

47. 1st Case We calculate the measure x of AC
Finding a side when we know two angles and a side
We calculate the measure x of AC

48. How to: Place Measurement x over sin known angle Equal to
Measurement known side over sin of known angle
Cross multiply and divide to find unknown measurement
Calculate.

49. 2nd Case
Finding an angle when we know two sides and the opposite angle to one of these two sides
We calculate the measure of angle B

50. Make sure you have opposite angles and side measurements
Make sure you have opposite angles and side measurements. Remember total inside angles must equal 180º

51. How to calculate if need to find an angle:
Place side measurement known over sin of angle we wish to know
Equal to
side measurement over sin angle we know
Cross multiply and divide to find x
To calculate angle –sin x = angle. Don’t forget unit i.e.º

52. Class work and homework
Math 3000, page 190, number 1 a and b – we will do altogether
Class work: page 190, number 1 c-f and number 2
Finish all above work tonight

53. The sine of an obtuse angle
The trigonometric functions (sine, cosine, etc.) are defined in a right triangle in terms of an acute angle.  What, then, shall we mean by the sine of an obtuse angle ABC?

54. The sine of an obtuse angle is defined to be the sine of its supplement.
How to find the measure of the degree of an obtuse angle:
Follow the procedure you have learned so far, then subtract that angle from 180º

55. 10 cm
22º
18.6 cm

56. Class work – page 190 #4,6