1 Trigonometry - Basic Ideas

The origins of trigonometry lie in the efforts of the ancient Babylonian and Greek as-tronomers and astrologers to understand the motions of the sun and the visible planets.Later trigonometry became an essential tool for navigation and surveying. Today trigonom-etry has immense applications in nearly every area of technology. For instance, trigonom-etry is essential to our understanding of sound, how to record it, and how to transmit it.Compression techniques for both audio and video signals rely on being able to representsignals in terms of trigonometric functions. We simply could not we do much of what wedo without trigonometry.

Trigonometry can be thought of in two ways , first as the study of the relationships betweenthe lengths of the sides of a triangle as a function of its angles or second as the mathematicsof a circle. Both perspectives are useful. We begin with a very basic question. What is anangle? We all have ideas. Given two lines that intersect, we can speak of the angle betweenthem as a method of measuring how close the two lines are to one another. How wouldwe define such a measure? We could define it as the number one would get by measuringthe separation with a protractor. But this depends on the precision of the protractor andalthough it is essential in practical situations it is not a method that can be incorporatedinto mathematical methods.

1.1 Angles

Given an arbitary line and a point O on the line, the point O divides the line into twohalf lines which start at O. These half lines are called rays. A second point, say P , onthe given line will determine one of the rays, which is denoted as

−−→OP. An angle consists of

two rays,−−→OP and

−−→OQ with the same starting point. Usually these rays are derived from

different lines. The job is to describe a method of measuring the separation between thetwo rays.

This is what is done. Let one of the rays say−−→OP correspond with the positive portion of

the real line and such that the point O is the origin (0, 0) of R2. Next consider a circle ofradius 1 centered at the origin. Let P be chosen so that P = (1, 0) and let Q be chosenso that Q = (x, y) is the intersection of the second ray

−−→OQ with the circle. The angle can

now be represented by the three points P,O, and Q as ∠POQ.

Starting at P the circle can be traversed in two ways to arrive at the point Q - in a counter-clockwise fashion or in a clockwise fashion. The corresponding arcs connecting P with Qcan be referred to as the counter-clockwise arc and the clockwise arc. The measurement ofthe angle determined by the two rays can then also be done in two ways. It can be measuredpositively by the length of the counter-clockwise arc or it can be measured negatively as the

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Figure 1: ∠POQ measured positively as the length of the connecting counter-clockwise arcor negatively as the length of the clockwise arc.

length of the clockwise arc. That is, measuring counter-clockwise gives a positive numberand measuring clockwise gives a negative number. This measurement according to thelength of arcs on the unit circle is called radian measure.

Figure 2: An angle with positive counterclockwise radian measure π2 and negative clockwise

radian measure −3π2

Since the circumference of the unit circle is 2π, the circumference of a quarter circle is π2 .

Thus if the point Q is located at the point (0, 1), angle ∠POQ is said to be a right angle,and its Its positive counterclockwise radian measure is π

2 . where as its negative measure asthe length of the connecting clockwise arc which is −3π

2 . If Q = (−1, 0), angle ∠POQ is180o and has positive radian measure π, since the connecting arc is a half circle. In thiscase the clockwise arc is also a half circle, so that the negative radian measure is −π.

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Figure 3: an angle with positive radian measure π and negative clockwise radian measure−π

1.1.1 Conversion between degrees and radians

When we measure angles with degrees, think of the circumference of the unit circle asdivided into 360 equal segments. Each of these is further divided into 60 equal subunitscalled minutes and each minute is further divided into 60 subunits called seconds. Thisis what we understand. Using ratios we can develop a method for converting from degreemeasure to radian measure and conversely from radian measure to degree measure. Nowit is clear that the length of a circular arc divided by the circumference is the same nomatter what style of measurement we decide to use. We could measure in kilometers orin microns; the ratio of arc length to circumference would be the same. So, if we measureby degrees or by radians the fraction arc length

circumference is the same. Thus, if d is the degreemeasurement of an angle and r is the radian measure, we know that

d

360=

r

2π

Thus, for instance, converting an angle of 60o to radian measure, gives

60360

=r

2π.

Solving for r we have r = π3 .

1.1.2 Generalized notion of angle

The description of an angle as two rays joined at their initial points has useful generaliza-tions. What we have shown is that an angle ∠POQ can be identified with two arcs on theunit circle, a counter-clockwise arc and a clockwise arc and that the length of these arcs arethe radian measures of the angle. The length of the counter-clockwise arc is representedas a positive number and the length of the clockwise arc as a negative number. Thus any

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angle as we have described it gives us two numbers a number θ where 0 ≤ θ ≤ 2π repre-senting the length of a counter-clockwise arc and the number −(2π − θ) = θ− 2π which isthe length of the corresponding clockwise arc. Note, that we allow for the possibility of anarc zero length or of length 2π.

We know that an angle can be identified with two arcs on the unit circle. In many appli-cations it is useful to alter slightly the definition of what we mean by the word “angle”,to refer simply to an arc on the circle starting at P = (0, 1) and continuing to a point Q -in either a counter-clockwise or a clockwise direction. Of course any such arc determinesan angle previously understood so this is not a big change. It does however now offer theadvantage that the word angle now refers to only one entity - either a positively measuredcounter-clockwise arc or the negatively measured clockwise arc. In this way an angle cannow be represented uniquely by the measure of its length as an arc on the circle; that is asa number θ, where −2π ≤ θ ≤ 2π. This is the first generalization.

The second generalization has arisen from the need to represent multiple revolutions aboutthe circle which are followed by an angle as we presently know it. For instance 2 fullrevolutions in the counter-clockwise direction followed by an angle of 90o is represented bythe length of the path traversed by a point which starts a P and takes two full revolutionsfollowed by a quarter revolution. The path length is 2π + 2π + π

2 = 4π + π2 .

Figure 4: an angle consisting of one counter-clockwise complete revolution plus the poritonfrom P to Q

In this way any real number, whether positive or negative, can be used to represent sucha path. How is this done? Let x be a real number; divide x by 2π to find how manyfull revolutions of the circle it represents. The remainder after the division represents thelength of the arc that is tacked onto the sequence of full circle revolutions. For instance ifx = 45 radians, dividing by 2π gives

452π≈ 7.163 = 7 + 0.163 radians.

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Multiplying both ends of the equation by 2π gives

45 ≈ 7× 2π + 2π × 0.163 ≈ 7(2π) + 1.024

Thus 45 in radian measure represents aproximately 7 counter-clockwise revolutions plus apartial revolution of 1.024 radians.

1.2 Right triangle trigonometry

We begin the study of trigonometric functions in a special setting in which the anglesconsidered are less than 90o and form one of the angles of right triangle. Later this canbe easily generalized to the case of a generalized angle whose measure is an arbitrary realnumber. Given a right triangle ∆POQ whose right angle is at the vertex P and ∠POQ = θ,the various trigonometric functions describe relationships between the sides of the triangleas a function of θ. The hypotenuse is the side OQ; the side PQ which opposite the angel∠POQ is said to be the opposite side, and the side OP is called the adjacent side. Seefigure 5. If θ = 0. we assume Q = P and the triangle degenerates to the line segment OP.Similarly if θ = π

2 we assume that O = P and the triangle degenerates to the line segmentOQ. We also assume a notational device; if A and B are any two points the length of theline segment AB is denoted AB.

Figure 5: right triangle

Definition 1 Given a right triangle as above, with the above conventions the trigonometricfunctions, sine, cosine, tangent, cotangent, secant, cosecant, are defined as follows for0 ≤ θ ≤ π

2 .

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1. sine of θ : sin θ =PQ

OQ=

opposite

hypotenuse

2. cosine of θ : cos θ =OP

OQ=

adjacent

hypotenuse

3. tangent of θ : tan θ =PQ

OP=

opposite

adjacent, for θ 6= π

2 .

4. cotangent θ : cot θ =OP

PQ=

adjacent

opposite, for θ 6= 0

5. secant of θ : sec θ =OQ

OP=

hypotenuse

adjacent, for θ 6= π

2 .

6. cosecant of θ : csc θ =OQ

PQ=

hypotenuse

opposite, for θ 6= 0.

Note that for θ = π2 the triangle ∆POQ collapses to the straight line segment OQ so

that OP = 0, and therefore the restrictions on the definition of tangent and secant arenecessary. Similarly if θ = 0, the triangle collapses to the line segment OP, so that PQbecomes zero, and the restrictions on the definitions of cotangent and cosecant are againnecessary.

The trigonometric functions of angles, θ equal to 0, π2 , π

4 , π3 and π

6 can be easily calculateby considering the triangle ∆POQ and assigning appropriate lengths to various sides.Specifically, we have the following.

Figure 6: If θ = π3 , then ∆POQ is half an equilateral triangle

Remark 2 Basic Results

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1. if θ = 0, then PQ = 0, so sin 0 = 0 and tan 0 = 0.

2. if θ = π2 , then OP = 0, so cos π

2 = 0 and cot π2 = 0.

3. if θ = π4 = 45o, then ∆POQ is a right triangle and we choose OP = 1 = PQ, then

the hypotenuse OQ has length√

2. And, knowing all the dimensions of the triangle,the values of all the trigonometric functions can be calculated. In particular

(a) tan π4 = 1 = cot π

4 ,

(b) sin π4 = 1√

2= cos π

4

4. if θ = π3 = 600, then ∆POQ is half of an equilateral triangle, and if we chose OQ to

be of length 2, it follows that OP = 1, and by the pythagorean theorem PQ =√

3.See figure 6.

Following is an example of how these ideas may be immediately applied.

Figure 7: Find the height of the CN tower

Example 3 A George Brown College student with surveying equipment has determinedthat the shadow of the CN tower downtown Toronto is 3, 200 feet long at a time that theelevation of the sun was 29.5o. How high is the tower?

Solution: Let h stand for the height of the tower. Then we must have that tan 29.5 =h

3210. Solving for h gives,

h = 3210× tan 29.5 = 3210× 0.56577 = 1816.1.

Not a bad result. The true height is 1,815 feet four inches.

In the next section we show how to extend the definitions of the trigonometric functionsto the case in which the angle θ can be any real number.

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1.3 Trigonometric functions of arbitrary angles

For the definitions of trigonometric functions given in the previous section, the variousratios of one side to another of the triangle ∆POQ were independent of the size of thetriangle. Thus had we wished we could have chosen a triangle of some standard size. Thisis what we will do here.

Figure 8: Trigonometry definitions with standard triangle

For the following definitions assume that the triangle is such that the hypotenuse OQ haslength 1. See figure 8. Note the similarity of 8 to that of figure 1, the difference being thatthe point P has now been moved along the x axis so that there is a right angle at P .

Since the hypotenuse OQ has length 1, the definitions of the trigonometric functions becomesimpler. The sine of the angle ∠POQ is simply the y coordinate of the point Q, the cosineis the x coordinate, the tangent is the ratio

y

x, and so on.

This definition allows a simple extension that lets us define trigonometric functions of anarbitrary angle. As we have seen, an arbitrary angle θ, which may include several completerotations in either in the counter-clockwise or the clockwise directions, is determined pointQ = (x, y) anywhere on the unit circle. We then define the trigonometric functions of suchan angle in the same way. That is,

Definition 4 In relation to figure 9, the trigonometric functions of an arbitrary angle θare defined as follows.

1. sin θ = y

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Figure 9: Trigonometric definitions for an arbitrary angle

2. cos θ = x

3. tan θ =y

xfor x 6= 0

4. cot θ =x

yfor y 6= 0

5. sec θ =1x

for x 6= 0

6. csc θ =1y

for y 6= 0

We remark that if Q lies in the first quadrant then it’s coordinates range from 0 to 1,whereas if Q lies in the second quadrant, the x coordinate ranges from 0 to −1 while the ycoordinate is still positive and ranges from 0 to 1. In the third quadrant, both coordinatesrange from 0 to −1, and in the fourth quadrant x ranges from 0 to 1 and y ranges from 0to −1.

1.4 Graphs of the trigonometric functions

The graphs of the trigonometric functions are an important tool in understanding whatgoes on. The graphs can be visualized by slowly considering what happens to the x and ycoordinates of the point Q = (x, y) as it moves around the circle. In the case of the sinefunction, the y coordinate of Q begins at y = 0 when the corresponding angle θ = 0 and

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Figure 10: graph of sin θ for 0 ≤ θ ≤ 2π

Figure 11: Showing sin(−θ) = −y = − sin θ

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Q corresponds to the point P = (1, 0). As θ increases, the y coordinate increases until itreaches a value of y = 1 and the point Q corresponds to the point (0, 1) on the y axis.Then as θ moves from 90o or π

2 radians to 180o or π radians, the y coordinate decreases toa value of 0. Finally, as θ moves from π to 360o or 2π radians, the y coordinate behavesin the same fashion as before, except that now the values are negative. The graph for0 ≤ θ ≤ 2π can be seen in figure 10.

Figure 12: Graph of sin θ for −2π ≤ θ ≤ 2π

Now what about negative values of θ? Suppose the angle θ is measured positively in thecounter-clockwise direction and is determined by the point Q = (x, y). Then −θ is measuredin the clockwise direction. It is determined by the point Q̃ = (x,−y); see fiigure 11. Thisallows us to expand the graph to include negative values of θ; see figure 12

Further, for angles greater than 2π or less that −2π, the values of the trigonometric func-tions of such angles are again determined by the coordinates of a point on the unit circle.For instance, if θ = n × 2π + ω, then θ consists of n complete revolutions followed byan angle ω in the range 0 ≤ ω ≤ 2π. Definition 4 tells us that sin θ is defined as the ycoordinate of the point Q on the unit circle which is now determined by the angle ω.

Saying this in a slightly different way, the definitions of the trigonometric functions of anarbitrary angle first expressed in Definition 4 may be rephrased as follows .

Definition 5 Given an angle ω, where 0 ≤ ω ≤ 2π, a positive integer n and a trigonomet-ric function f

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1. if θ = 2nπ + ω, then f(θ) = f(ω)

2. if θ = −2nπ − ω, then f(θ) = f(−ω).

Figure 13: Graph of the sine function on [−6π, 6π]

In other words, in each interval [2π, 4π], [4π, 6π], · · · , [2nπ, 2(n+1)π], · · · , the 6 trigonomet-ric functions replicate the behavior which they demonstrate on the interval [0, 2π]. Likewiseon each of the intervals [−4π,−2π], [−6π,−4π], · · · , [−2(n + 1π,−nπ] · · · , they replicatethe behavior demonstrated on [−2π, 0]. In particular the graph of the sine function on theinterval [−6π, 6π] is as seen in figure 13.

1.4.1 The graph of the cosine function

As with the sine function to visualize the graph of the cosine function we need to examinethe values of the x coordinate of the point Q = (x, y) as it moves about the unit circle.Let θ let be the counter-clockwise angle made by the ray

−−→OQ with the positive x axis. If

θ = 0, the point Q corresponds to the point (1, 0), so that cos 0 = 1. As Q moves counterclockwise, the x coordinate decreases until it becomes zero when θ = π

2 = 90o. Thus,

cosπ

2= 0. Then as θ goes from π

2 to π, the x coordinate goes from 0 to −1, so that

cos π = −1. Similar analysis shows that as θ goes from π to3π

2, cos θ goes from −1 to 0,

and as θ goes then from 3π2 to 2π, cos θ goes form 0 to plus 1.

Finally repeating the analysis for negative angles −2π ≤ θ ≤ 0, we arrive at the graphin figure 14. As with the sine function , the behavior of the cosine function on intervals[2nπ, 2(n + 1)π] and [−2(n + 1)π,−2nπ] replicates the behavior on [0, 2π] and [−2π, 0].Figure 15 shows the graph of the cosine on the interval [−6π, 6π]

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Figure 14: Graph of the cosine on [−2π.2π]

1.4.2 Beginning Identities

In figure 16 we show the a portion of graph of the cosine function superimposed on aportion of the graph of the sine function. Notice that the red colored graph of the cosineis simply a translation to the left by π

2 of the green colored graph of the sine function.Similarly, the sine function is a translation to the right by π

2 of the cosine function. Wecan express these observations analytically as follows.

Result 6 Translation Identities

1. cos θ = sin(θ + π2 )

2. sin θ = cos(θ − π2 )

Examining the graph of the sine function as in figure 13, we see that the reflection aboutthe y axis of the portion of the sine graph for positive θ gives the negative of the graph ofthe sine for negative θ. In other words, to obtain the portion graph of the sine for negativeθ, take the portion for positive θ, flip it about the y axis and then reflect the result aboutthe x axis.

As for the cosine function, examining figure 14 we see that the reflection about the y axisof the the portion of the graph for θ > 0 corresponds precisely with the graph of the cosinefor negative θ. Said in another way, to obtain the portion of the graph of the cosine fornegative θ, take the portion for positive θ, and simply flip it about the y axis.

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Figure 15: Graph of the cosine on the interval[−6π, 6π]

Figure 16: Graph of the cosine in red is a translate of the graph of the sine in green andvice-versa

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These ideas can be expressed as follows.

Result 7 : Reflection Identities

1. For any θ, sin(−θ) = − sin θ

2. For any angle θ, cos(−θ) = cos θ

Reflecting back on the definition of the sine and cosine functions, an angle θ is determinedby a point Q = (x, y) on the unit circle so that sin θ = y and cos θ = x. However letsconsider the triangle ∆POQ of figure 1 where 0 ≤ θ ≤ π

2 . Since this is a right triangle wethen know that x2 + y2 = 1 by the Pythagorean theorem. For other values of θ a similartriangle can be constructed . The result is that sin2 θ + cos2 θ = 1

Result 8 : Pythagorean Identity For any angle θ, sin2 θ + cos2 θ = 1

Again focusing on the definitions in terms of the coordinates of the point Q = (x, y)

determined by an angle θ, we know thattan θ =y

x=

sin θ

cos θand cot θ =

x

y=

cosθ

sin θ.

Similarly sec θ =1x

=1

cos θand csc θ =

1y

=1

sin θ. This is summarized in the following

statement.

Result 9 : Basic Identities

1. tan θ =sin θ

cos θ

2. cot θ =cos θ

sin θ

3. tan θ =1

cot θ

4. sec θ =1

cos θ

5. csc θ =1

sin θ

1.4.3 Graph of tangent, cotangent, secant, and cosecant

The graphs of the remaining trigonometric functions can be derived from those of sine and

cosine. For instance, for the tangent function, since tan θ =sin θ

cos θ, we know:

• if θ = 0, then sin θ = 0 and therefore tan θ = 0;

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Figure 17: Graph of the tangent on the interval (−π2 , π

2 )

Figure 18: Graph of the tangent on an extended domain

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• if 0 < θ < π2 , then both sin θ and zcos θ are positive, but as θ gets close to π

2 , cos θgets close to zero whereas sin θ gets close to 1. The result is that tan θ approaches+∞;

• if −π2 < θ < 0 a similar analysis tells us that tan θ is negative and approaches −∞

as θ approaches −π2 .

The graph of the tangent function over the interval for −π2 θ < π

2 is shown in diagram 17.Note that the graph becomes asymptotic to the vertical lines through −π

2 and π2 .

A similar analysis will show that the graph of the tangent will be the same on all ad-joining intervals of length π - namely the intervals (π

2 , 3π2 ), (3π

2 , 5π2 ) · · · and the intervals

(−3π2 ,−π

2 ), (−5π2 ,−3π

2 ), · · · . The graph of the tangent over an extended domain is shownin diagram 18. Observe that the cos θ = 0 if θ is a multiple of π

2 ; that is cos nπ2 = 0

for all integers n, whether they be positive or negative. Consequently at all such pointsθ, tan θ is not defined and the graph of the tangent is asymptotic to the vertical linethrough θ = nπ

2 .

The graphs of the cotangent, the secant, and the cosecant can be derived similarly fromthose of the sine and the cosine. The analysis is omitted and left as an exercise for thestudent . The graphs of the cotangent, secant and cosecant are shown in figures 19, 20, and21 respctively. Observe that just as the graph of the cosine is a translation of the graphof the sine, so the graph of secant is a translation of the graph of cosecant. Also observethat the distinct curves described by both the secant and the cosecant are tangent to thehorizontal lines through (0, 1) and (0,−1), namely the lines y = 1 and y = −1.

Figure 19: Graph of the cotangent

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Figure 20: Graph of the secant

Figure 21: Graph of the cosecant

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