Download ppt - Math 416 Trigonometry. Time Frame 1) Pythagoras 1) Pythagoras 2) Triangle Structure 2) Triangle Structure 3) Trig Ratios 3) Trig Ratios 4) Trig Calculators

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


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…


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


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.


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


x

12

40°

Step #2 – Determine the equation

X = sin 40°

12

Step #3: Cross multiply (if necessary)

x = 12 Sin 40°


x

12

40°

Step #4 If the unknown is isolated (by itself) solve… if not

divide then solve

x = 7.71


More PracticeMore Practice

x

39°

11

x = sin 39°

11

x = 11 sin 39°

x = 6.92


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


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


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?


Find the SideFind the Side

x

61°6

x = Tan 61°

6

x = 6 tan 61°

x = 10.82



7

θ

31

sin θ = 7

31

sin θ ( 7 )

31

θ= 13°



5

θ

7

cos θ = 5

7

θ = 44°



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