1. Math 416
Trigonometry

2. Time Frame 1) Pythagoras 2) Triangle Structure 3) Trig Ratios
4) Trig Calculators
5) Trig Calculations
6) Finding the angle
7) Triangle Constructions
8) Word Problems

3. Right Angle Triangles
The next section will deal exclusively with right angle triangles. We recall
Pythagoras
x2 = y2 + z2
Angle Sum
θ+ β + = 180° β x y θ z

4. Pythagoras Example Do Stencil #1  72 = x2 + 32 49 = x2 + 9 40 = x2 7

5. Triangle Structure
We all need to agree on what we are talking about. Consider A
<BAC = <A
<ABC = <B
<BCA = <C
AB = c opposite
BC = a opposite
CA = b opposite b c C B a
Do Stencil #2 

6. Trig Ratios
When we consider the similarity of right angle triangles as long as we ignore decimal angles there are only 45 right angle triangles.
Consider the angles
90° – 1° – 89°
90° – 2°– 88°
90° – 3° – 87° …
90° – 45° - 45°
Then we start over

7. Trig Ratios
From ancient times, people have looked at the ratios within right angles triangles
First in tables
Now stored in calculators
We need to define the parts of a right angle triangle
Two types of definitions

8. Definitions
Absolute – never changes
Relative – involves the position

9. Absolute vs Relative
Now we can define absolutely the hypotenuse as the side opposite the right angle (longest side). In this example it is side AC or b. A
Now “relative to angle θ” we define side AB or c as the opposite side
Now “relative to angle β” we define side BC or a as the opposite side β b c θ B C a

10. Absolute Vs. Relative
Now “relative to angle θ” we define side BC or a as the adjacent side
Now “relative to angle β” we define side AB or c as the adjacent side

11. Labeling the Triangle Hence with respect to θ
Now we define the three main trig ratios…
Hyp
Opp θ
Adj

12. Trig Ratios
The sine of an angle is defined as the ratio of the opposite to the hypotenuse. Thus Sin θ= Opp
Hyp
The cosine of an angle is defined as the ratio of the adjacent to the hypotenuse. Thus Cos θ= Adj

13. Trig Ratios
The tangent of an angle is defined as the ratio of the opposite to the adjacent. Thus Tan θ = Opp
Adj

14. SOH – CAH - TOA
You may of heard the acronym SOH – CAH – TOA or SOCK – A – TOA
Sin Opp
Hyp
Cos Adj
Tan Opp
Adj

15. Old Harry And His Old Aunt
There is another acronym… old Harry and his old aunt
Sin Opp
Hyp
Cos Adj
Tan Opp
Adj
Use the acronym that you can remember

16. Example Sin A = 15 39 Sin C = 36 39 Consider A Cos A = 36 39
Cos C = 15 39 39 36 B C 15
Tan A = 15 36
Tan C = 36 15

17. Trig Calculator
Now note the table for the assignment is as follows (question #3). For example B 40 24 A C 32
# Angle Sin Cos Tan Angle Sin Cos Tan 32 40 24 40 32 40
Eg B 24 40 32 24 C 24 32

18. Trig Calculator
We note that these ratios are stored by angle albeit as decimals in a calculator
Note first and foremost your calculators
IT MUST BE IN DEGREES
Make sure you find your DRG (Degree – Radian – Gradients)

19. Trig Calculator Hence if θ = 54° then to 4 decimal places Sin 54° =
0.8090
Cos 54° =
0.5878
Tan 54° =
1.3764
Do Stencil #3 

20. Question #4 The table required for #4 is as follows Example θ = 37°
# Sin Cos Tan Eg
0.6018
0.7986
0.7536

21. Trig Calculations
There are three basic type of questions. We will focus on the Sine ratio (like question #5) but the techniques are the same for all trig ratio problems.

22. Trig Calculations Solve for x Consider Use the angle given to you!12
Step #1: Determine the Trig Ratio involved with respect to the angle x
40°
12 = hypotenuse, x = opp Thus, SINE

23. Step #2 – Determine the equation
Trig Calculations
Step #2 – Determine the equation 12
X = sin 40° 12 x
40°
Step #3: Cross multiply (if necessary)
x = 12 Sin 40°

24. Trig Calculations
Step #4 If the unknown is isolated (by itself) solve… if not divide then solve 12
x = 7.71 x
40°

25. Trig Calculations More Practice x = sin 39° 11 x = 11 sin 39° 11

26. Trig Calculations Even More Practice 11 = sin 42° x 11 = x sin 42° x
Divided both sides by sin 42° or 0.67

27. Trig Calculations Even More Practice 9 = sin 73° x x x = 9 . sin 73° 9

28. Finding the Angle Up until now we have the angle get the ratio
Now we need to go the other way
Given the ratio, give the angle
Eg. The buttons we are looking for are the inverse sine (sin -1)
Inverse cosine (cos -1)
Inverse tangent (tan -1)
Find it on your calculator

29. Examples of Finding the Angle
Find the angle
Sin θ = 5 16
θ= sin -1 ( 5 )
θ= 18°
(no decimals) 16 5 θ

30. Another Example
Find the angle
sin θ = 7 31
θ= 13° 31 7 θ

31. Other Examples
Now all the Trig Calculations can follow these procedures
x = cos 25° 15
x = 13.59 15
25° x
Sin, Cos or Tan?

32. Another Example Find the Side x = Tan 61° 6 x = 6 tan 61° x = 10.82 x

33. Another Example
Find the angle
sin θ = 7 31
sin θ ( 7 )
θ= 13° 31 7 θ

34. Another Example
Find the angle
cos θ = 5 7
θ = 44° 7 θ 5

35. Another Example
Find the angle
tan θ = 18 5
θ = 74° 18 θ 5

36. Completing the Triangle
Now using our knowledge we can complete triangles
Draw this triangle and another one right below… fill out missing info
76° x
5 = cos 14° x
x = 5.15
y = tan 14° 5
y = 1.25 y
14° 5