Taming Trigonometry

8/13/2019 Taming Trigonometry

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Rick Bowman The Algebra Toolbox 2005

Taming

Trigonometry!Skip the intro & take mestraight to the examples!

Id like to do the intro!


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Trigonometry is that branch of mathematics thatdeals with sides and angles in triangles.

In Years 9 and 10, we work with

Right Angled Triangles.


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1. Naming the angles

In triangles, the threeangles are usually namedusing capital letters, for

example A, B and C.

A

BCIt is important to notethat in this diagram,

Angle C (also called

C)is 90.The side opposite the90 is always the longestside in the triangle (calledthe hypotenuse)

Hypotenuse AB


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2. Naming the sides

We have alreadyseen how angles arelabelled (with capital

letters). Now weintroduce a methodof labelling thesides oppositethose angles (withsmall letters).

Its very simple!Pretend thetriangle is a yard.

If we stand in corner C and

look out across thetriangular yard,

C

A

B

c

the fence we can seefurthest away is called c.


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C

A

B

c

In the same way, if we

stood in corner B

andlooked out across thetriangular yard then thefence furthest awaywould be called b.

b

And likewise if we stood atcorner A, then the sidefurthest away would be called

a.

a


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Exercise 2 questions to test yourknowledge!

P

R

M

1. Label all thesides using lowercase letters

ANSWER

2. Which side is thehypotenuse?

ANSWER


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P

R

M

pm

r

How did yougo??

BACK


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P

R

M

pm

r

The hypotenuse isthe longest side,which is alwaysopposite the 90.

Here, the hypotenuse issidep.

Next section

You could also call it RM


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2. Naming the sides (continued)

We have already metthe hypotenuse. Itsthe side that sits

opposite the 90

The other two sides are called adjacent andopposite, but these names are interchangeable,depending on which angle we are working with.

A

C B


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A

C B

Suppose we decideour reference angle

(i.e. the angle wherewe are standinginside the yard) is B

This is often representedby the Greek letter (called theta) rather thandrawing a little person!

The side furthest away from

is called theOPPOSITE, and...

OPP

OSITE

The side next to (other than the Hypotenuse)iscalled the ADJACENT.

ADJACENT


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A

C B

Suppose instead, ourreference angle was in

corner A

This would make thenames of the other twosides change.

The opposite side would now be here,OPPOSITE

because thisside is the furthest away from reference angle .

The hypotenuse stays where it is (its still thelongest side!) This means that the remaining sidemust be the adjacent.

ADJACENT

NOTE!! The reference angle is never the 90, butalways one of the other 2 remaining (acute) angles


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Now tryWorksheet #1


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3. Similar Triangles

In this triangle ABC,the sides are 3cm, 4cmand 5cm. The angle at Bis .

Now suppose all thesides are doubled. 10cm

8cm

6cm

Because this is a magnification

or enlargement of the smallertriangle, its shapewill still bethe same, and so its angles willremain the same.

5cm

4cm

3cm

A

BC


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Rick BowmanThe Algebra Toolbox

2005

5cm

4cm

3cm

A

BC

Now, in the smaller triangle using as our reference angle, find whichsides are opposite and adjacent

and calculate the ratio AdjacentOpposite

A

O

4

30.75

Opp

Adj

10cm

8cm

6cm

A

BC

Now do the same thing for the bigtriangle

A

O

8

60.75

Opp

AdjNow calculate the ratio forboth triangles. Hypotenuse

Opposite

You should get 3/5 or 0.6 for BOTH

same!


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2005

Summary so far

(1) When all 3 of a triangles sides are multiplied (or divided)by the same number, the shape of the triangle doesnt

change (only the size). All angles stay the same size.Two triangles related this way (where one is anenlargement or blow-up of the other) are called SIMILAR.

(2) Whenever we have two similar triangles, and calculatethe fractions (or ratios) of matching pairs of sides like wedid on the previous slide, we always get the same result.

So the size of the triangle doesnt matter, only the shape!

So, suppose we have a triangle like the one onthe next slide


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2005

8cm 17cm

15cm

Lets say this trianglessides are doubled, halvedand tripled to get threenew triangles drawn below.

(not to scale, of course!)

16cm34cm

30cmx2

4cm 8.5cm

7.5cm2 24cm 51cm

45cm

x3


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If we now take all 4 triangles and for each one, pick thesame 2 sides and work out the ratio of their lengths, wellget the same result! Try it for

Hypotenuse

Adjacent

Original

Orig x2

Orig

2Orig x3

17

15

H

A0.88

34

30

0.88

0.88

0.88

5.8

5.7

51

45

Now try it forHypotenuse

Opposite Did you get 0.47 for all four ?

And finally AdjacentOpposite

Hopefully you got 0.53 for allfour ?

Opp Hyp

Adj


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4. New names for the ratios!

HypotenuseOpposite is called SINE (abbreviated to sin)

Hypotenuse

Adjacent is called COSINE (cos)

Adjacent

Opposite is called TANGENT (tan)

SOHCAHTOA


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2005

Now tryWorksheet #2


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4. Finding a side

Now well actually use trigonometryto find unknown sides, but first wemay need a brief refresher courseon 2 important Algebra skills!

Do the refresher

Skip the refresher

SOHCAHTOA


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Skill #1 Solving an equation when the unknownis in the topof a fraction

Example..

Solve 35

x

Method..multiplyboth sides by the bottomnumber, 5

5

x= 3x 5 x 5

x= 15answer

The 5s cancelout


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Skill #2 Solving an equation when the unknownis in the bottomof a fraction

Example..

Solve 35

x

Method..interchange the xand the 3. Keep thetop number (5) where it is

x3

5 This enables you tomake xthe subject

in one stepx= 1.67answer

Previous slide

Next section


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4. Finding a side Example 1

32

23mxmFind xin this shape.

STEP 1 Look at your reference angle (32). Carefullylabel the hyp, then opp and adj.

HypOpp

Adj

Remind me how

STEP 2 Choose which two sides you are concerned with.They are labelled one has 23m, the other xm.

These are the Opp and Hyp

We do not bother with Adj.


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Remember Slide 18? We use ONE of these formulas now.

HypotenuseOpposite is called SINE (abbreviated to sin)

Hypotenuse

Adjacent is called COSINE (cos)

Adjacent

Opposite is called TANGENT (tan)

Which do you think we use?

If you said sin youd be right, becausewere dealing with Opp and Hyp


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STEP 3 Set up this equation:

32

23mxm

sin 32 =Hyp

Opp

Now get sin 32 from

calculator,23

5299.0 x

Solve using Algebra Skill #1 (Slide 21: xis in the top)

x= 0.5299 23

x= 12.19(to 2 dec places)

and replaceOpp with xand Hyp with23

Next example


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2005

A

C B

Suppose we decideour reference angle

(i.e. the angle wherewe are standinginside the yard) is B

This is often represented

by the Greek letter (called theta) rather thandrawing a little person!

The side furthest away from is called theOPPOSITE, and...

OPPOSITE

The side next to (other than the Hypotenuse)iscalled the ADJACENT.

ADJACENT


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2005

A

C B

Suppose instead, ourreference angle was in

corner AThis would make thenames of the other twosides change.

The opposite side would now be here,OPPOSITE

because thisside is the furthest away from reference angle .

The hypotenuse stays where it is (its still the

longest side!) This means that the remaining sidemust be the adjacent.

ADJAC

ENT

NOTE!! The reference angle is never the 90, butalways one of the other 2 remaining (acute) angles

Back to Example 1


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Find yin thistriangle. 5.2 cm

ycm 28

STEP 1 Look at your reference angle (28). Label thehyp, then opp and adj.

HypAdj

Opp

STEP 2 Opp is not labelled, so we dont use it. Hyp and Adjare the ones we use. This is COS

SOHCAHTOA


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STEP 3

5.2 cm

ycm 28

Hyp

Adj28cos

y

2.58829.0

8829.0

2.5y

using Algebra Skill #2 (Slide 22:xis in the bottom)

89.5y to 2 dec places


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Looking at the angle

41, OPP and ADJ arelablled so this timewell use TAN

SOHCAHTOA

Adj

Opp41tan

208693.0

p

p = 17.39

20m41

p mFindp

8693.020p Skill #1: Slide 21


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Now tryWorksheet #3


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5. Finding an angle Example 1

Find the size of angle , to the

nearest minute

10cm

8cm

STEP 1Look at your referenceangle (). Label the hyp,then opp and adj.

Hyp

Adj

Opp

Which two are labelled? Opp(8cm) and Adj(10cm),which means well use Tan

STEP 2

Adj

Opp =tan

10

8

=tan

10

8tan=

1

= 3840

Key this in as

2ndTAN (8/10)Once you have decimaldegrees on your screen, hit2ndANGLE DMSto getdegrees and minutes (Texascalculators)


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5. Finding an angle Example 2

Find the size of angle , to the

nearest minute

STEP 1Look at your referenceangle (). Label the hyp,then opp and adj.

Hyp

Adj Opp

Which two are labelled? Hyp(15cm) and Adj(11cm),which means well use Cos

STEP 2

Hyp

Adj =cos

15

11

=cos

-

15

11cos=

1

= 4250

Once you have decimaldegrees on your screen, hit2ndANGLE DMSto getdegrees and minutes (Texascalculators)

11m

15m

Key this in as

2ndCOS (11/15)


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Now tryWorksheet #4

Taming Trigonometry

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