Post on 04-Jun-2018
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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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Rick Bowman The Algebra Toolbox 2005
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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Rick Bowman The Algebra Toolbox 2005
P
R
M
pm
r
How did yougo??
BACK
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Rick Bowman The Algebra Toolbox 2005
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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Rick Bowman The Algebra Toolbox 2005
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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Rick Bowman The Algebra Toolbox 2005
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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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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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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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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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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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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4. Finding a side Example 2
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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4. Finding a side Example 3
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