1. Metric Relations in Right Triangles

2. By drawing the altitude from the right angle of a right triangle, three similar right triangles are formed

3. Geometric Properties
Using the lengths of the corresponding sides of the triangles formed, we can determine the ratios and from this determine certain geometric properties

4. Property 1
In a right triangle the length of the leg of a right triangle is the geometric mean between the length of its projection on the hypotenuse
A.)

5. In easiest terms, a leg squared is equal to the hypotenuse multiplied by the leg’s projection on the hypotenuse.

6. Property 2:
The square of the altitude is equal to one part of the hypotenuse multiplied by the other

7. Property 3:
In a right triangle, the product of the length of the hypotenuse and its corresponding altitude is equal to the product of the lengths of the legs.

8. Class Work and Homework
Visions P. 116 # 9
Visions P. 115 # 2-8, 10, 13, 15, 19
Math 3000 P. 217 # 1, 3, 4

9. Trigonometry

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

11. 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

12. 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)

13. 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 the hypotenuse or opposite
Ex: Side BC is adjacent to m<C
Side AB is adjacent to m<A

14. Example: Fill in the side that corresponds to the following questions:
Hypotenuse: _________________
Opposite m<A: _________________
Adjacent m<A: _________________
Opposite m<C: __________________
Adjacent m<C: __________________

15. 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

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

18. Example:
60° 2 1
30°

19. 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

20. 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

21. Hand outs 1 and 2: Trigonometric Ratios and Calculator
Homework
Hand outs 1 and 2: Trigonometric Ratios and Calculator
Math 3000 pages 228, 229 # 2,3,4,5
P. 182 # 1
P. 184 # 2

22. Finding Missing Sides Using
Trigonometric Ratios

23. Finding the Missing Side of a Right Triangle
In order to find a missing side, you will need two pieces of information:
An angle
One side length

24. 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

25. 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

26. 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

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

28. Example In the following triangle, find the value of side x
(Remember to use SOH-CAH-TOA)!!!

29. Example In the following triangle, find the value of side y
(Remember to use SOH-CAH-TOA)!!!

30. Example
Find the value of x

31. Class work and homework
Finding missing sides
math 3000: page 229 # 6,7
Hand out

32. Finding Missing Angles using Trigonometry Ratios

33. 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

34. 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

35. 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

36. Find the missing Angle
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

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

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

39. Class work and homework
Hand out Find the missing angles using trig ratios Math 3000 page 231 # 8,9

40. Math 3000 pg 232 # 10-15 x=10.4 cm; y=3.0 cm; z=9.0 cm; t=5.2 cm
ab=10cos64 = 4.38 cm; bc=10sin64= 8.99 cm; area: cm squared
H=50tan40 = 42 m
Length of shadow = 50tan30 = 28.9 m
X=65tan54 degrees = 89.5 m
B). c=50 degrees, ab=4,6, ac=3.9
mc=60 decrees, ab=5.2, ac=3
Bc=10, angle b=52.1 degrees, angle c=36.9 degrees
Ac=12, angle b=67.4, angle c=22.6
Ab=15, angle b=28.1 degrees, angle c=61.9 degrees

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. Sine Law

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

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

45. 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

46. 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.

47. 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

48. 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

49. 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

50. 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?

51. 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

52. 10 cm
22º
18.6 cm

53. Class assignment
Find all missing side lengths and angles and math 3000 page 232 # 10-15

54. There are three formulas to find the area of any triangle
Area of a Triangle
There are three formulas to find the area of any triangle

55. The first of the three you are already aware of:
A = b x h 2

56. Area of a Triangle knowing Two Sides and the Angle in Between
Area = ac ● sinB 2
Area = ab ● sinC
Area = bc ● sinA A c b B C a

57. These are the 3 formulas that involve the sine of an angle and the two sides that contain the angle

58. Example: - Find the area of the following triangle Area = ac●sinB 2 Area = 6 ●12sin(36.3°) Area = 21.3cm²
6cm
36.3°
12cm

59. Hero’s Formula Where ‘p’ = HALF of the perimeter Example:
8cm
6cm
12cm

60. Step 1: Label the sides of the triangle if relevant Step 2: State what information we know Step 3: Select a theorem and find the area

61. In easiest terms, you need two side lengths and the angle in between them to find the area of any triangle

62. - Find the area of the following triangle
Example:
- Find the area of the following triangle
Area = ac●sinB 2
Area = 6 ●12sin(36.3°)
Area = 21.3cm²
6cm
36.3°
12cm

63. Area of a Triangle Knowing Two Angles and One Side
Step 1: Draw the Altitude AD Step 2: Find mAD Step 3: Find mBD Step 4: Find mDC Step 5: Add mBD and mDC Step 6: Use any formula you wish to find the total area A
10cm 55 25 B C D

64. Class work and homework
Math 3000 pg 233 act 1 and 2, and pg 235 numbers 1-5, 7 and hand outs