Trig

Trig is also covered inAppendix C of the text.

1 SOH CAH TOAThese relations were first in-troduced for a right angledtriangle to relate the angle,itsopposite and adjacent sidesand the hypotenuse.

sin(θ) =opp

hyp

cos(θ) =adj

hyp

tan(θ) =opp

adj

Example: In a right-angledtriangle with a 30◦ angle oppo-site a 4 metre side, what arethe other sides and angles?The next concept to tackle

is the angle measurements indegrees versus radians.

Unlike degrees, radian an-2

gle measures are a natural partof the mathematics.

2 “What’s a ra-dian?”• Draw a circle with an x-yaxes in the centre

• Cut a string the same lengthas the diameter.

• Cut that string in half so nowyou have a string that is oneradius long.

• If you lay your one radiuslong string down on the

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circle, the angle subtendedby that string is a one radianangle. This translates toapproximately 57.32 degreesregardless of the size of thecircle you drew. Neat eh?

How many radius stringlengths it takes to go com-pletely around the circle?If you measured perfectly

you would get 2π times. Thatis, a full circle is 2π radiansaround.We also know a full circle

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is 360 degrees around so360◦ = 2π radians

You can use this equation toconvert to and from degrees toradians.

We will always talk aboutangles as measured in radiansbut you need to convert todegrees to converse with theoutside world. You can’t tellthe cab driver make a π

2 radianturn at the next light.

Now trig is always first in-troduced to you in high school

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as a triangle thing. (You re-member soh cah toa?) Thetruth is, trigonometry is veryconnected to the circle, a shapeancient people considered per-fect because it had no cornersor edges, no beginning or end.Say our circle has a radius of 1unit. We can then call it a unitcircle. Any point on that circlehas 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 andsin θ functions come from.

Cast Rule?We will now see that every-

thing falls out of that unitcircle. Take a 90 degree ( orπ2) angle for example. Thecoordinates on the unit cir-cle are (0, 1), therefore the

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cos(90◦) = 0, sin(90◦) = 1. Try iton your calculator.• What about the angle 270◦ =3π2 ?

• 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:tan(θ) =

sin (θ)

cos(θ)

sec(θ) =1

cos(θ)

csc(θ) =1

sin(θ)

cot(θ) =cos(θ)

sin(θ)=

1

tan(θ)So, once we know the cos(θ)and sin(θ) from the x and ycoordinates, we can calculatethe other 4 trig values.Example: What is the

cot(−270◦)?Also, do you remember

the equation for a circle with9

centre (0, 0) and with radius r?Yes, x2 + y2 = r2.

We will learn more aboutcircles later.

Well, in our case the unitcircle has radius 1 so theequation of the unit circle isx2+y2 = 1.Noting that x = cos(θ)and y = sin(θ) we can see that

cos2(θ) + sin2(θ) = 1

That is the “father trigono-metric identity”. Divide thatwhole identity equation by

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cos2(θ) to getcos2(θ)

cos2(θ)+sin2(θ)

cos2(θ)=

1

cos2(θ)

1 + tan2(θ) = sec2(θ)

or divide that whole identityequation by sin2(θ) and get

cos2(θ)

sin2(θ)+sin2(θ)

sin2(θ)=

1

sin2(θ)

cot2(θ) + 1 = csc2(θ)

This is how we derive the threeidentities which you shouldmemorize.

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 tosin(θ), tan(θ) to sec(θ) and cot(θ)to csc(θ). This is a pattern thatwill reoccur when we learn thederivatives of trig functions.But wait, you say. I can

only get the x and y coordi-nates for 4 points, where theunit circle crosses the axes.What about other angles like45◦? First memorize these tworight angled triangles.

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121

21

1 12

32

45

45 30

60

Notice their hypoteni areboth 1 unit long. Redraw themwith the radian angle measuresso we can use them in the nextfew examples.

ExampleQ? Find cos(π6) exactly.A. We need to find the

x-coordinate that corresponds13

to the angle of π6 = 30◦. Us-

ing the second triangle above,we can fit it into the unit cir-cle. Now you see why thehypotenuse is one unit, so fitsexactly into our unit circle.

We can see that the x-coordinate is

√32 . Therefore

cos(π6) =√32 .

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You can confirm the answerto the above Example on a cal-culator by getting a decimalfor

√32 and a decimal for cos(π6).

Remember to have your cal-culator in radian mode. If thetwo decimal approximationsare the same, then you can behappy with your exact result.

ExampleQ? Evaluate sec(225◦)

exactly.15

3 Trig GraphsThe unit circle also helpsus understand why the trigfunctions have the graphs theydo. The x and y coordinatesboth oscillate between +1 and−1 as the angle increases fromzero. So do the graphs.y = cos(x)

-1

-0.5

0

0.5

1

-4 -2 2 4x

y = sin(x)

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-1

-0.5

0

0.5

1

-4 -2 2 4x

Show using the unit circlethat cos(x) is even and sin(x)is odd. Notice these factscorrespond to the graphs.

We will learn to graphmore trig functions later

The other memorization Iwant you to do is these doubleangle 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 formu-lae will help you later in higherorder calculus courses. Andthe answers in the back of thebook are at times the result ofa major simplification usingthese trig equations.Now some exercises to

solidify your new trig knowl-edge.

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Do exercises.1 to18, 28 to 32 in Appendix Cof the text. Submit numbers4,10,12,14,18,28,30,32 neatlyat the beginning of class.

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