Upload
kathryn-austin
View
225
Download
0
Embed Size (px)
Citation preview
Math 416Math 416
Trigonometry Trigonometry
Time FrameTime Frame
1) Pythagoras1) Pythagoras 2) Triangle Structure2) Triangle Structure 3) Trig Ratios3) Trig Ratios 4) Trig Calculators4) Trig Calculators 5) Trig Calculations5) Trig Calculations 6) Finding the angle6) Finding the angle 7) Triangle Constructions7) Triangle Constructions 8) Word Problems 8) Word Problems
Right Angle TrianglesRight Angle Triangles
The next section will deal exclusively The next section will deal exclusively with right angle triangles. We recallwith right angle triangles. We recall
yx
θz
β
Pythagoras
x2 = y2 + z2
Angle Sum
θ+ β + = 180°
Pythagoras Pythagoras
ExampleExample
x
3
7
72 = x2 + 32
49 = x2 + 9
40 = x2
6.32 = x
x = 6.32
Do Stencil #1
Triangle StructureTriangle Structure
We all need to agree on what we are We all need to agree on what we are talking about. Considertalking about. Consider
b
B C
A
c
a
<BAC = <A
<ABC = <B
<BCA = <C
AB = c opposite
BC = a opposite
CA = b opposite Do Stencil #2
Trig RatiosTrig Ratios
When we consider the similarity of right When we consider the similarity of right angle triangles as long as we ignore angle triangles as long as we ignore decimal angles there are only 45 right decimal angles there are only 45 right angle triangles. angle triangles.
Consider the anglesConsider the angles 90° – 1° – 89°90° – 1° – 89° 90° – 2°– 88°90° – 2°– 88° 90° – 3° – 87° … 90° – 3° – 87° … 90° – 45° - 45°90° – 45° - 45° Then we start overThen we start over
Trig RatiosTrig Ratios
From ancient times, people have From ancient times, people have looked at the ratios within right looked at the ratios within right angles trianglesangles triangles
First in tablesFirst in tables Now stored in calculatorsNow stored in calculators We need to define the parts of a right We need to define the parts of a right
angle triangleangle triangle Two types of definitionsTwo types of definitions
DefinitionsDefinitions
Absolute – never changesAbsolute – never changes Relative – involves the positionRelative – involves the position
Absolute vs RelativeAbsolute vs Relative
c
Cθ
a
β
b
B
A
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.
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
Absolute Vs. RelativeAbsolute Vs. Relative
Now “relative to angle Now “relative to angle θθ” we define ” we define side BC or a as the adjacent sideside BC or a as the adjacent side
Now “relative to angle Now “relative to angle ββ” we define ” we define side AB or c as the adjacent sideside AB or c as the adjacent side
Labeling the TriangleLabeling the Triangle
Hence with respect to Hence with respect to θθ
Hyp
θ
Adj
Opp
Now we define the three main trig ratios…
Trig RatiosTrig Ratios
The sine of an angle is defined as the The sine of an angle is defined as the ratio of the opposite to the ratio of the opposite to the hypotenuse. Thus Sin hypotenuse. Thus Sin θθ= = OppOpp
HypHyp The cosine of an angle is defined as The cosine of an angle is defined as
the ratio of the adjacent to the the ratio of the adjacent to the hypotenuse. Thus Cos hypotenuse. Thus Cos θθ= = AdjAdj
HypHyp
Trig RatiosTrig Ratios
The tangent of an angle is defined as The tangent of an angle is defined as the ratio of the opposite to the the ratio of the opposite to the adjacent. Thus Tan adjacent. Thus Tan θθ = = OppOpp
AdjAdj
SOH – CAH - TOASOH – CAH - TOA
You may of heard the acronym SOH – You may of heard the acronym SOH – CAH – TOA or SOCK – A – TOACAH – TOA or SOCK – A – TOA
Sin Sin OppOpp HypHyp Cos Cos AdjAdj HypHyp Tan Tan OppOpp AdjAdj
Old Harry And His Old AuntOld Harry And His Old Aunt
There is another acronym… old Harry and There is another acronym… old Harry and his old aunthis old aunt
Sin Sin OppOpp HypHyp Cos Cos AdjAdj HypHyp Tan Tan OppOpp AdjAdj Use the acronym that you can rememberUse the acronym that you can remember
ExampleExample
ConsiderConsider
39
C15
36
Sin A = 15
39A
BTan C = 36
15
Cos C = 15
39
Cos A = 36
39
Tan A = 15
36
Sin C = 36
39
Trig CalculatorTrig Calculator Now note the table for the Now note the table for the
assignment is as follows (question assignment is as follows (question #3). For example#3). For example
40
C32
24
B
A# Angle Sin Cos Tan Angle Sin Cos Tan
Eg B 32
4024
40
32
24
C 24
40
32
4024
32
Trig CalculatorTrig Calculator
We note that these ratios are stored We note that these ratios are stored by angle albeit as decimals in a by angle albeit as decimals in a calculatorcalculator
Note first and foremost your Note first and foremost your calculatorscalculators
IT MUST BE IN DEGREESIT MUST BE IN DEGREES Make sure you find your DRG Make sure you find your DRG
(Degree – Radian – Gradients) (Degree – Radian – Gradients)
Trig CalculatorTrig Calculator
Hence if Hence if θθ = 54° then to 4 decimal = 54° then to 4 decimal placesplaces
Sin 54° = Sin 54° = 0.8090
•Cos 54° = 0.5878
•Tan 54° = 1.3764
Do Stencil #3
Question #4Question #4
The table required for #4 is as The table required for #4 is as followsfollows
Example Example θθ = 37° = 37° # # SinSin CosCos TanTan EgEg 0.6018 0.75360.7986
Trig CalculationsTrig Calculations
There are three basic type of There are three basic type of questions. We will focus on the Sine questions. We will focus on the Sine ratio (like question #5) but the ratio (like question #5) but the techniques are the same for all trig techniques are the same for all trig ratio problems. ratio problems.
Trig CalculationsTrig Calculations
ConsiderConsider
x
12
40°
Solve for x
Use the angle given to you!
Step #1: Determine the Trig Ratio involved with respect to the angle
12 = hypotenuse, x = opp Thus, SINE
Trig CalculationsTrig Calculations
x
12
40°
Step #2 – Determine the equation
X = sin 40°
12
Step #3: Cross multiply (if necessary)
x = 12 Sin 40°
Trig CalculationsTrig Calculations
x
12
40°
Step #4 If the unknown is isolated (by itself) solve… if not
divide then solve
x = 7.71
Trig CalculationsTrig Calculations
More PracticeMore Practice
x
39°
11
x = sin 39°
11
x = 11 sin 39°
x = 6.92
Trig CalculationsTrig Calculations
Even More PracticeEven More Practice
11
42°
x
11 = sin 42°
x
11 = x sin 42°
Divided both sides by sin 42° or 0.67
x = 16.44
Trig CalculationsTrig Calculations
Even More PracticeEven More Practice
9
73°
x9 = sin 73°
xx = 9 .
sin 73°
x = 9.41
Finding the AngleFinding the Angle
Up until now we have the angle get the Up until now we have the angle get the ratioratio
Now we need to go the other wayNow we need to go the other way Given the ratio, give the angleGiven the ratio, give the angle Eg. The buttons we are looking for are the Eg. The buttons we are looking for are the
inverse sine (sin inverse sine (sin -1-1)) Inverse cosine (cos Inverse cosine (cos -1-1)) Inverse tangent (tan Inverse tangent (tan -1-1)) Find it on your calculatorFind it on your calculator
Examples of Finding the AngleExamples of Finding the Angle
Find the angleFind the angle
5
θ
16
Sin θ = 5
16
θ= sin -1 ( 5 )
16
θ= 18°
(no decimals)
Another Example Another Example
Find the angleFind the angle
7
θ
31
sin θ = 7
31
θ= 13°
Other ExamplesOther Examples
Now all the Trig Calculations can Now all the Trig Calculations can follow these proceduresfollow these procedures
25°
15
x = cos 25°
15
x = 13.59
xSin, Cos or Tan?
Another Example Another Example
Find the SideFind the Side
x
61°6
x = Tan 61°
6
x = 6 tan 61°
x = 10.82
Another Example Another Example
Find the angleFind the angle
7
θ
31
sin θ = 7
31
sin θ ( 7 )
31
θ= 13°
Another Example Another Example
Find the angleFind the angle
5
θ
7
cos θ = 5
7
θ = 44°
Another Example Another Example
Find the angleFind the angle
5
θ
18
tan θ = 18
5
θ = 74°
Completing the TriangleCompleting the Triangle
Now using our knowledge we can Now using our knowledge we can complete trianglescomplete triangles
5
14°
76°
y
x
Draw this triangle and another one right below… fill out missing info
5 = cos 14°
x
x = 5.15
y = tan 14°
5
y = 1.25