Download 1 SOH CAH TOA - RIT

Trig
Trig is also covered in
Appendix C of the text.
1 SOH CAH TOA
These relations were first introduced for a right angled
triangle to relate the angle,its
opposite and adjacent sides
and the hypotenuse.
opp
sin(θ) =
hyp
adj
cos(θ) =
hyp
opp
tan(θ) =
adj
Example: In a right-angled
triangle with a 30◦ angle opposite a 4 metre side, what are
the other sides and angles?
The next concept to tackle
is the angle measurements in
degrees versus radians.
Unlike degrees, radian an2
gle measures are a natural part
of the mathematics.
2 “What’s a radian?”
Draw a circle with an x-y
axes in the centre
• Cut a string the same length
as the diameter.
• Cut that string in half so now
you have a string that is one
radius long.
• If you lay your one radius
long string down on the
•
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circle, the angle subtended
by that string is a one radian
angle. This translates to
approximately 57.32 degrees
regardless of the size of the
circle you drew. Neat eh?
How many radius string
lengths it takes to go completely around the circle?
If you measured perfectly
you would get 2π times. That
is, a full circle is 2π radians
around.
We also know a full circle
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is 360 degrees around so
360◦ = 2π radians
You can use this equation to
convert to and from degrees to
radians.
We will always talk about
angles as measured in radians
but you need to convert to
degrees to converse with the
outside world. You can’t tell
the cab driver make a π2 radian
turn at the next light.
Now trig is always first introduced to you in high school
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as a triangle thing. (You remember soh cah toa?) The
truth is, trigonometry is very
connected to the circle, a shape
ancient people considered perfect because it had no corners
or edges, no beginning or end.
Say our circle has a radius of 1
unit. We can then call it a unit
circle. Any point on that circle
has x and y coordinates.
The x coordinate of any
point on the unit circle is cos θ,
the y coordinate is sin θ.
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That’s where the cos θ and
sin θ functions come from.
Cast Rule?
We will now see that everything falls out of that unit
circle. Take a 90 degree ( or
π ) angle for example. The
2
coordinates on the unit circle are (0, 1), therefore the
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cos(90◦) = 0, sin(90◦) = 1.
Try it
on your calculator.
• What about the angle 270◦ =
3π ?
2
•
Consider the angle −90◦
There are six trig values,
cos(θ), sin (θ) , tan(θ), sec(θ), csc(θ)
and cot(θ).
They are related by the
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equations:
sin (θ)
tan(θ) =
cos(θ)
1
sec(θ) =
cos(θ)
1
csc(θ) =
sin(θ)
cos(θ)
1
cot(θ) =
=
sin(θ) tan(θ)
So, once we know the cos(θ)
and sin(θ) from the x and y
coordinates, we can calculate
the other 4 trig values.
Example: What is the
cot(−270◦)?
Also, do you remember
the equation for a circle with
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centre (0, 0) and with radius r?
Yes, x2 + y2 = r2.
We will learn more about
circles later.
Well, in our case the unit
circle has radius 1 so the
equation of the unit circle is
x2 +y 2 = 1. Noting that x = cos(θ)
and y = sin(θ) we can see that
cos2(θ) + sin2(θ) = 1
That is the “father trigonometric identity”. Divide that
whole identity equation by
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cos2(θ) to get
cos2(θ) sin2(θ)
1
+ 2
=
2
cos (θ) cos (θ)
cos2(θ)
1 + tan2(θ) = sec2(θ)
or divide that whole identity
equation by sin2(θ) and get
1
cos2(θ) sin2(θ)
+ 2
=
2
sin (θ) sin (θ)
sin2(θ)
cot2(θ) + 1 = csc2(θ)
This is how we derive the three
identities which you should
memorize.
cos2(θ) + sin2(θ) = 1
1 + tan2(θ) = sec2(θ)
cot2(θ) + 1 = csc2(θ)
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See how cos(θ) is related to
sin(θ), tan(θ) to sec(θ) and cot(θ)
to csc(θ). This is a pattern that
will reoccur when we learn the
derivatives of trig functions.
But wait, you say. I can
only get the x and y coordinates for 4 points, where the
unit circle crosses the axes.
What about other angles like
45◦? First memorize these two
right angled triangles.
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45(
1
1
2
45(
1
60(1
2
30(
1
2
3
2
Notice their hypoteni are
both 1 unit long. Redraw them
with the radian angle measures
so we can use them in the next
few examples.
Example
Q?
Find cos( π6 ) exactly.
A.
We need to find the
x-coordinate that corresponds
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to the angle of π6 = 30◦. Using the second triangle above,
we can fit it into the unit circle. Now you see why the
hypotenuse is one unit, so fits
exactly into our unit circle.
We can see√ that the x3
coordinate
is
2 . Therefore
√
cos( π6 ) = 23 .
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You can confirm the answer
to the above Example on a calculator
by getting a decimal
√
for 23 and a decimal for cos( π6 ).
Remember to have your calculator in radian mode. If the
two decimal approximations
are the same, then you can be
happy with your exact result.
Example
Q?
Evaluate sec(225◦)
exactly.
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3 Trig Graphs
The unit circle also helps
us understand why the trig
functions have the graphs they
do. The x and y coordinates
both oscillate between +1 and
−1 as the angle increases from
zero. So do the graphs.
y = cos(x)
1
0.5
-4
-2
0
-0.5
-1
y = sin(x)
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2 x
4
1
0.5
-4
-2
0
2 x
4
-0.5
-1
Show using the unit circle
that cos(x) is even and sin(x)
is odd. Notice these facts
correspond to the graphs.
We will learn to graph
more trig functions later
The other memorization I
want you to do is these double
angle formulae.
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sin(2x) = 2 sin(x) cos(x)
cos(2x) = cos2(x) − sin2(x)
= 2 cos2(x) − 1
= 1 − 2 sin2(x)
These double angle formulae will help you later in higher
order calculus courses. And
the answers in the back of the
book are at times the result of
a major simplification using
these trig equations.
Now some exercises to
solidify your new trig knowledge.
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Do exercises.1 to
18, 28 to 32 in Appendix C
of the text. Submit numbers
4,10,12,14,18,28,30,32 neatly
at the beginning of class.
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