Review of Trigonometry for Calculus · 2002-08-18 · Review of Trigonometry for Calculus 1 U n i v ersit a s S a sk atchew n e n s i s DEO ET PAT-RIÆ 2002 Doug MacLean Review of

Review of Trigonometry for Calculus 1Universitas

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Review of Trigonometry for Calculus

“Trigon” =triangle +“metry”=measurement=Trigonometry

so Trigonometry got its name as the science of measuring triangles.

When one first meets the trigonometric functions, they are presented in the context of ratios of sides of a right-angledtriangle, where a2 + o2 = h2:

a

o

h

α

We have

sinα = oh= opposite

hypotenuse,

cosα = ah= adjacent

hypotenuse, and

tanα = oa= opposite

adjacent,

which is often remembered with the “sohcahtoa” rule. If h = 1, we have sinα = o, cosα = a, so (sinα)2 + (cosα)2 = 1,so we know that the point (cosα, sinα) lies on the unit circle.

2 Review of Trigonometry for CalculusUniversitas

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The Unit Circle:In Calculus, most references to the trigonometric functions are based on the unit circle, x2+y2 = 1. Points on this circledetermine angles measured from the point (0,1) on the x-axis, where the counter- clockwise direction is considered tobe positive.

Units of Angular MeasurementThe most natural unit of measurement for angles in Geometry is the right angle . The revolution is used in the

study of rotary motion, and is what the “r” stands for in “rpm”s. The degree , 1/90 of a right angle, was probably first

adopted for navigational purposes. The mil , 1/1600 of a right angle, is used by the military. However, the basic unitof measurement for angles in Calculus is the radian .

Definition: A radian is the angle subtended by a circular arc on a circle whose length equals the radius of thecircle. Thus, on the unit circle an angle whose size is one radian subtends a circular arc on the unit circle whose lengthis exactly one.

1 radian

x

y

(0,0) (1,0)

Figure 1.


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Radian measure and degreesSince the circumference of a circle is 2π times its radius, we have

2π radians = 360◦ = 4 right angles,

so

1 radian = 3602π

◦= 180

π

◦= 4 right angles

2π= 2π

right angles

or

1◦ = 2π360

radians = π180

radians = 190

right angles

In high school trigonometry, the trigonometric functions are used to solve problems concerning triangles and relatedgeometric figures. In the Calculus, the trigonometric functions are used in the analysis of rotating bodies. It turns outthat the degree, the unit of measurement of angles adopted by the Babylonians over 4,000 years ago, is not particularlywell adapted to the analysis of jet engines, radar systems and CAT scanners.

The radian is, because The sine and cosine functions live on the unit circle!If θ is a number, then cosθ and sinθ are defined to be the x- and y- coordinate, respectively, of the point on the unitcircle obtained by measuring off the angle θ (in radians!) from the point (0,1). If θ is positive, the angle is measuredoff in the counter-clockwise direction, and if θ is negative it is measured off in the clockwise direction. For an animatedinteractive look at these two functions, take a look at the applet Sine and Cosine Functions


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x

y

(0,0) cos

sinθ

θ

θ

Figure 2.

The other trigonometric functions are now defined in terms of the first two:

tanθ = sinθcosθ

, cotθ = cosθsinθ

, secθ = 1cosθ

, cscθ = 1sinθ

.


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Fundamental Angles of the First Quadrant:

There are three acute angles for which the trigonometric function values are known and must be memorized by thestudent of Calculus. They are

(in radians) π6 , π

4 , and π3 ,

(in degrees) 30◦, 45◦, and 60◦,(in right angles) 1/3, 1/2, and 2/3.

In addition, the values of the trig functions for the angles 0 and π2 must be known. The following tables show how they

may be easily constructed, if one can count from zero to four. The first table is a template, the second shows how it maybe filled in, and the third contains the arithmetical simplifications of the values.

Template:

θ(radians) 0 π6

π4

π3

π2

θ(degrees) 0 30 45 60 90

θ(right angles) 0 13

12

23 1

sinθ√

2

√2

√2

√2

√2

cosθ√

2

√2

√2

√2

√2


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Fill in the Blanks:

θ(radians) 0 π6

π4

π3

π2

θ(degrees) 0 30 45 60 90


12

23 1

sinθ√

02

√1

2

√2

2

√3

2

√4

2

cosθ√

42

√3

2

√2

2

√1

2

√0

2

Simplify the Arithmetic:

θ(radians) 0 π6

π4

π3

π2

θ(degrees) 0 30 45 60 90


12

23 1

sinθ 0 12

√2

2

√3

2 1

cosθ 1√

32

√2

212 0


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Figure 3 shows these values on the first quadrant of the unit circle.

x

y

(0,1)( , )

( , )

( , )

(1,0)

— —

— —

— —

—

— —

—

2 22 2

2 2

1 32 2

3 1

ππ

π

π

√√ √

√

——

—

—

23

4

6

0

Figure 3.


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Moving Beyond the First QuadrantThese values may now be used to find the values of the trig functions at the other basic angles in the other three quadrantsof unit circle. The same numerical values will appear, with the possible addition of minus signs. The following table givesthe values, and the diagram displays them. The student should be able to reproduce them instantaneously! To do this, itwill be necessary to be completely comfortable with the following identities, all of which are obvious from the symmetryof the unit circle:

x

y

θ

−θθ+π

π−θ

π/2−θ

Figure 4.


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2002 Doug MacLeansin(π − θ) ≡ sinθ, cos(π − θ) ≡ − cosθ

sin(θ +π) ≡ − sinθ, cos(θ +π) ≡ − cosθ

sin(−θ) ≡ − sinθ, cos(−θ) ≡ cosθ

sin(π2− θ

)≡ cosθ, cos

(π2− θ

)≡ sinθ

θ 0 π6

π4

π3

π2

2π3

3π4

5π6 π 7π

65π4

4π3

3π2

5π3

7π4

11π6

sinθ 0 12

√2

2

√3

2 1√

32

√2

212 0 −1

2 −√

22 −

√3

2 −1 −√

32 −

√2

2 −12

cosθ 1√

32

√2

212 0 −1

2 −√

22 −

√3

2 −1 −√

32 −

√2

2 −12 0 1

2

√2

2

√3

2


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Figure 5 is left blank for the student to fill in:

x

y

(1,0)

( , )

( , )

( , )(0,1)

( , )

( , )

( , )

(-1,0)

( , )

( , )

( , )

(0,-1)

( , )

( , )

( , )

——

——

————

——

——

— —

— —

— — — —

— —

— —

—

— —

—

2 2

2 2

2 2

1 3

2 2

3 1

ππ

π

π

π

π

π

π

ππ

π

π

π

π

π

√

√ √

√

——

—

—

—

—

—

——

—

—

—

—

—

23

4

6

3

4

6

23

4

6

3

4

6

2

3

5

4

5

7

35

7

11

0

Figure 5.


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Periodicity All six trig functions have period 2π , and two of them, tan and cot have period π :

sin(θ + 2π) ≡ sin(θ)cos(θ + 2π) ≡ cos(θ)

tan(θ +π) ≡ tan(θ)cot(θ +π) ≡ cot(θ)

sec(θ + 2π) ≡ sec(θ)csc(θ + 2π) ≡ csc(θ)


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Identities of the sine and cosine functionsThe identity

sin2 θ + cos2 θ ≡ 1

is obvious as a result of our use of the unit circle. It really should be written as

(sinθ)2 + (cosθ)2 ≡ 1

but centuries of tradition have developed the confusing convention of writing sin2 θ for the square of sinθ. This identityleads to a number of other important identities and formulas:

tan2 θ ≡ sec2 θ − 1

sec2 θ ≡ 1+ tan2 θ + 1

cot2 θ ≡ csc2 θ − 1

csc2 θ ≡ 1+ cot2 θsinθ = ±

√1− cos2 θ

In addition to this fundamental knowledge, the student should be completely comfortable in deriving the trig identitieswhich result from the fundamental identities for the sines and cosines of sums and differences of angles. First we need:


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The Law of Cosines: c2 = a2 + b2 − 2ab cosγ

a

c

bw

z

γπ−γ

We have w = b sin(π − γ) = b sinγ, and z = b cos(π − γ) = −b cosγ, so

c2 = w2 + (a+ z)2 = (b sinγ)2 + (a− b cosγ)2 = b2 sin2 γ + a2 − 2ab cosγ + b2(cosγ)2 = a2 + b2 − 2ab cosγ

Next we compute c2 slightly differently:

a

c

b h

x yα β

x = b cosα, y = a cosβ, h = b sinα = a sinβ,x2 = b2 − h2, y2 = a2 − h2, so

c2 = (x +y)2 = x2 + 2xy +y2 =b2 − h2 + 2(b cosα)(a cosβ)+ a2 − h2 =a2 + b2 − 2h2 + 2ab cosα cosβ =a2 + b2 − 2ab sinα sinβ+ 2ab cosα cosβ =a2 + b2 − 2ab(sinα sinβ− cosα cosβ)

so cosγ = sinα sinβ− cosα cosβ = − cos(π − γ) = − cos(α+ β).Therefore

cos(α+ β) = cosα cosβ− sinα sinβ


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We collect the angle sum and difference formulae:

sin(α+ β) ≡ sinα cosβ+ sinβ cosα (1)

sin(α− β) ≡ sinα cosβ− sinβ cosα (2)

cos(α+ β) ≡ cosα cosβ− sinα sinβ (3)

cos(α− β) ≡ cosα cosβ+ sinα sinβ (4)

If we add (1) and (2) and divide by 2, we get

sinα cosβ ≡ 12(sin(α+ β)+ sin(α− β))

If we add (3) and (4) and divide by 2,we get

cosα cosβ ≡ 12(cos(α+ β)+ cos(α− β))

and if we subtract (3) from (4) we get

sinα sinβ ≡ 12(cos(α− β)− cos(α+ β))


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Double Angle Formulae:If we let β = α in (1) and (3) and divide by 2, we get:

sin 2α ≡ 2 sinα cosα (5)

cos 2α ≡ cos2α− sin2α = 2 cos2α− 1 = 1− 2 sin2α (6)

(6) leads to the two identities:

cos2α ≡ 1+ cos 2α2

(7)

sin2α ≡ 1− cos 2α2

(8)

which in turn lead to the formulas

cosα = ±√

1+ cos 2α2

sinα = ±√

1− cos 2α2

.


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These in turn lead to the

Half-Angle Formulae:

cosα2

= ±√

1+ cosα2

(9)

sinα2

= ±√

1− cosα2

(10)

The above identities may be used to compute the exact values of trig functions at many other angles, such as π8 = 1

2π4

and π12 , but in practice one usually uses a calculator or computer to get extremely accurate values of the trig functions.


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Identities of the Other Four Trigonometric FunctionsThese may all be derived from the preceding:For example,

tan(α+ β) ≡ sin(α+ β)cos(α+ β) ≡

sinα cosβ+ sinβ cosαcosα cosβ− sinα sinβ

≡sinα cosβ+ sinβ cosα

cosα cosβcosα cosβ− sinα sinβ

cosα cosβ≡ tanα+ tanβ

1− tanα tanβ,

tan(α+ π2) ≡ sin(α+ π

2 )cos(α+ π

2 )≡ cosα− sinα

≡ −1tanα

(thus the formula for slopes of perpendicluar lines).

tan(α− β) ≡ sin(α+ β)cos(α+ β) ≡

tanα+ tanβ1+ tanα tanβ

,

tan(2α) ≡ 2 tanα1− tan2α

,

and tanα2=

sinα2

cosα2

=±√

1− cosα2

±√

1+ cosα2

= ±1− cosα1+ cosα


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Inverse Trigonometric Functions

Definition: If h is a real number, the inverse sine or Arcsin of h is that number between −π/2 and π/2whose sine is h.

This is often called the Primary angle whose sine is h. It may be found geometrically by drawing the horizontal liney = h and observing the points where it intersects the unit circle. If there are two such points, the one on the rightdetermines the Primary Angle . The point on the left determines another angle whose sine is also h; this angle is

called the Secondary Angle . There are, of course, infinitely many other angles whose sine is h, they may all beobtained by adding integer multiples of 2π to the Primary or Secondary Angles. Geometrically, this is the same as goingcompletely around the unit circle a number of times and ending up at the same point.

Definition: If k is a real number, the inverse cosine or Arccos of k is that number between 0 and π whosecosine is k.

This is often called the Primary angle whose cosine is k. It may be found geometrically by drawing the verticalline x = k and observing the points where it intersects the unit circle. If there are two such points, the upper onedetermines the Primary Angle . The lower point determines another angle whose cosine is also k; this angle is called

the Secondary Angle . There are, of course, infinitely many other angles whose cosine is k, they may all be obtained byadding integer multiples of 2π to the Primary or Secondary Angles. Geometrically, this is the same as going completelyaround the unit circle a number of times and ending up at the same point.


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Java Applets:For an animated interactive look at these two functions, take a look at the applets ArcSine Applet and ArcCosine Applet

x

y

=Arcsin h

y=h

θ

θ

x

y

=Arcsin h

x=hθ

θ

Figure 6.


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AppendixIt is useful to know how the values of sinθ and cosθ for standard values of θ are derived, in addition to having memo-

rized them. First we begin with θ = π4= 45◦:

hh/2 h/2

x x

π/4 π/4

In a right angled isosceles triangle, the base angles are both equal toπ4

, and the hypotenuse h is equal to√x2 + x2 = x√2.

Therefore both the sine and cosine ofπ4

are equal toxh= xx√

2= 1√

2=

√2

2.


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Next we look at θ = π6= 30◦ and θ = π

3= 60◦ : we take an equilateral triangle whose sides are all of length x, and all

of whose angles areπ3

, and draw the perpendicular from the top vertex to the base, and in so doing bisecting the angle

at the top vertex.

π/3 π/3

π/6 π/6

x/2 x/2

x x

x

h

The perpendicular bisector has length h =√x2 −

(x2

)2

=√x2 − x

2

4= x

√1− 1

4= x

√34= x

√3

2

so we have

sinπ6=

x2

x= 1

2and cos

π6= x

√3

2

x=

√3

2.

Also, sinπ3= x

√3

2

x=

√3

2and cos

π3=

x2

x= 1

2.

Documents

Review of Trigonometry for Calculus · 2002-08-18 · Review of Trigonometry for Calculus 1 U n i v ersit a s S a sk atchew n e n s i s DEO ET PAT-RIÆ 2002 Doug MacLean Review of