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Review of Trigonometry for Calculus

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Review of Trigonometry for Calculus

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“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-angled triangle, where a2 + o2 = h2 : We have

sin α =

opposite o = , h hypotenuse

cos α =

adjacent a = , and h hypotenuse

tan α =

opposite o = , a adjacent

h o α a

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

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The Unit Circle:

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In Calculus, most references to the trigonometric functions are based on the unit circle, x 2 + y 2 = 1. Points on this circle determine angles measured from the point (0, 1) on the x-axis, where the counter- clockwise direction is considered to be positive.

Units of Angular Measurement The 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 unit of 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 the circle. Thus, on the unit circle an angle whose size is one radian subtends a circular arc on the unit circle whose length is exactly one. y

1 radian

(0,0)

Figure 1.

x (1,0)

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Radian measure and degrees

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Review of Trigonometry for Calculus

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Since the circumference of a circle is 2π times its radius, we have 2π radians = 360◦ = 4 right angles, so 1 radian = or 1◦ =

2 180 ◦ 4 right angles 360 ◦ = right angles = = 2π π 2π π

π 1 2π radians = radians = right angles 360 180 90

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

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

Review of Trigonometry for Calculus

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y

θ

sin θ

x (0,0)

cosθ

Figure 2. The other trigonometric functions are now defined in terms of the first two: tan θ =

sin θ , cos θ

cot θ =

cos θ , sin θ

sec θ =

1 , cos θ

csc θ =

1 . sin θ

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There are three acute angles for which the trigonometric function values are known and student of Calculus. They are π π (in radians) , , and π3 , 6 4 (in degrees) 30◦ , 45◦ , and 60◦ ,

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Fundamental Angles of the First Quadrant:

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must be memorized by the

(in right angles) 1/3, 1/2, and 2/3. π In addition, the values of the trig functions for the angles 0 and 2 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 may be filled in, and the third contains the arithmetical simplifications of the values.

Template:

θ(radians)

0

π 6

π 4

π 3

π 2

θ(degrees)

0

30

45

60

90

θ(right angles)

0

1 3

1 2

2 3

1

sin θ cos θ

√

√ 2

√

√ 2

√ 2

√ 2

√ 2

√ 2

√ 2

2 √

2

2

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Review of Trigonometry for Calculus Sa

Fill in the Blanks:

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θ(radians)

0

π 6

π 4

π 3

π 2

θ(degrees)

0

30

45

60

90

θ(right

0

1 3

1 2

2 3

1

√ 0 2 √ 4 2

√ 1 2 √ 3 2

√ 2 2 √ 2 2

√ 3 2 √ 1 2

√ 4 2 √ 0 2

angles)

sin θ cos θ

Simplify the Arithmetic: θ(radians)

0

π 6

π 4

π 3

π 2

θ(degrees)

0

30

45

60

90

θ(right

0

1 3

1 2

2 3

1

sin θ

0

1

1

√ 2 2 √ 2 2

√ 3 2

cos θ

1 2 √ 3 2

1 2

0

angles)

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

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y

(0,1) π — 2

— 1 , √—3 ) (— 2 2— — π — (√—2 ,√—2 ) 3 2 2— π — (√—3 ,1— ) 4

2 2

π — 6

0

Figure 3.

(1,0) x

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Moving Beyond the First Quadrant

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

y

π−θ

θ π/2−θ

x

θ+π

−θ

Figure 4.

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sin(π − θ) ≡ sin θ,

θ

0

sin θ

0

cos θ

1

π 6 1 2 √ 3 2

π 4 √ 2 2 √ 2 2

cos(π − θ) ≡ − cos θ

sin(θ + π ) ≡ − sin θ,

cos(θ + π ) ≡ − cos θ

sin(−θ) ≡ − sin θ,
 π sin − θ ≡ cos θ, 2

cos(−θ) ≡ cos θ
 π cos − θ ≡ sin θ 2

π 3 √ 3 2

π 2

1

2π 3 √ 3 2

1 2

0

− 12

3π 4 √ 2 2 √ − 22

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5π 6

π

7π 6

1 2 √ − 23

0

−2

−1 −

1

√

3 2

5π 4 √ 2 −2 √ − 22

4π 3 √ 3 −2

− 12

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Review of Trigonometry for Calculus

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3π 2

5π 3 √

−1 − 0

3 2

1 2

7π 4 √

11π 6

√

√ 3 2

−

2 2

2 2

1

−2

Review of Trigonometry for Calculus

y

(—, — ) 3—π 4

(—, —)

2π — 3

(0,1) π — 2

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

(— , —)

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π — 3

5—π 6

(-1,0)

(—, —)

— 1 √ 3) (—, — 2 2 — — 2 , √— 2) ( √— 2 2 π — — 3, — 1 ) √ (— 4 2 2 π — 6

π

0

π 7— 6

π 5— 4

(— , —) (— , —)

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π 4— 3

π 3— 2 Figure 5. (0,-1)

π 5— 3

π 7— 4

π 11— 6

x

(—, —)

( —, —) (— , —)

(1,0)

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

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Periodicity

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Review of Trigonometry for Calculus

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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)

Review of Trigonometry for Calculus Sa

Identities of the sine and cosine functions

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The identity sin2 θ + cos2 θ ≡ 1 is obvious as a result of our use of the unit circle. It really should be written as (sin θ)2 + (cos θ)2 ≡ 1 but centuries of tradition have developed the confusing convention of writing sin2 θ for the square of sin θ. This identity leads 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 θ  sin θ = ± 1 − cos2 θ In addition to this fundamental knowledge, the student should be completely comfortable in deriving the trig identities which 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:

c = a + b − 2ab cos γ 2

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2

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We have w = b sin(π − γ) = b sin γ, and z = b cos(π − γ) = −b cos γ, so c 2 = w 2 + (a + z)2 = (b sin γ)2 + (a − b cos γ)2 = b2 sin2 γ + a2 − 2ab cos γ + b2 (cos γ)2 = a2 + b2 − 2ab cos γ Next we compute c 2 slightly differently: z

x = b cos α, y = a cos β, h = b sin α = a sin β, x 2 = b2 − h2 , y 2 = a2 − h2 , so

π−γ w

2

2

a

b

c = (x + y) = x + 2xy + y = b2 − h2 + 2(b cos α)(a cos β) + a2 − h2 = a2 + b2 − 2h2 + 2ab cos α cos β = a2 + b2 − 2ab sin α sin β + 2ab cos α cos β = a2 + b2 − 2ab(sin α sin β − cos α cos β) 2

γ

2

c

so cos γ = sin α sin β − cos α cos β = − cos(π − γ) = − cos(α + β). Therefore cos(α + β) = cos α cos β − sin α sin β

b α

h

x

a y

c

β

Review of Trigonometry for Calculus Sa

We collect the angle sum and difference formulae:

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sin(α + β) ≡ sin α cos β + sin β cos α

(1)

sin(α − β) ≡ sin α cos β − sin β cos α

(2)

cos(α + β) ≡ cos α cos β − sin α sin β

(3)

cos(α − β) ≡ cos α cos β + sin α sin β

(4)

If we add (1) and (2) and divide by 2, we get sin α cos β ≡

1 (sin(α + β) + sin(α − β)) 2

If we add (3) and (4) and divide by 2,we get cos α cos β ≡

1 (cos(α + β) + cos(α − β)) 2

sin α sin β ≡

1 (cos(α − β) − cos(α + β)) 2

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

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Double Angle Formulae:

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Review of Trigonometry for Calculus

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

sin 2α ≡ 2 sin α cos α 2

2

(5) 2

2

cos 2α ≡ cos α − sin α = 2 cos α − 1 = 1 − 2 sin α

(6)

(6) leads to the two identities:

cos2 α ≡ sin2 α ≡ which in turn lead to the formulas  1 + cos 2α cos α = ± 2

1 + cos 2α 2 1 − cos 2α 2  sin α = ±

(7) (8)

1 − cos 2α . 2

Review of Trigonometry for Calculus Sa

These in turn lead to the

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Half-Angle Formulae:

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α cos 2

= ±

α sin 2

= ±



1 + cos α 2

(9)

1 − cos α 2

(10) π

1π

The above identities may be used to compute the exact values of trig functions at many other angles, such as 8 = 2 4 π and 12 , 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 Functions

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These may all be derived from the preceding: For example,

sin α cos β + sin β cos α sin(α + β) sin α cos β + sin β cos α tan α + tan β cos α cos β tan(α + β) ≡ ≡ ≡ , ≡ cos α cos β − sin α sin β cos(α + β) cos α cos β − sin α sin β 1 − tan α tan β cos α cos β tan(α +

sin(α + π2 ) π cos α −1 )≡ ≡ (thus the formula for slopes of perpendicluar lines). π ≡ 2 cos(α + 2 ) − sin α tan α

tan(α − β) ≡ tan(2α) ≡

sin(α + β) tan α + tan β ≡ , cos(α + β) 1 + tan α tan β

2 tan α , 1 − tan2 α



1 − cos α α ± sin 1 − cos α α 2 2 =  =± and tan = α 2 1 + cos α 1 + cos α cos ± 2 2

Review of Trigonometry for Calculus Sa

Inverse Trigonometric Functions

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Definition: whose sine is h.

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

This is often called the Primary angle whose sine is h. It may be found geometrically by drawing the horizontal line y = h and observing the points where it intersects the unit circle. If there are two such points, the one on the right determines 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 be obtained by adding integer multiples of 2π to the Primary or Secondary Angles. Geometrically, this is the same as going completely around the unit circle a number of times and ending up at the same point.

Definition: cosine is k.

If k is a real number, the inverse cosine or Arccos of k is that number between 0 and π whose

This is often called the Primary angle whose cosine is k. It may be found geometrically by drawing the vertical line x = k and observing the points where it intersects the unit circle. If there are two such points, the upper one determines 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 by adding integer multiples of 2π to the Primary or Secondary Angles. Geometrically, this is the same as going completely around the unit circle a number of times and ending up at the same point.

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Java Applets:

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Review of Trigonometry for Calculus

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

y

y y=h

θ

θ=Arcsin h

Figure 6.

x

θ

x=h

θ=Arcsin h

x

Review of Trigonometry for Calculus Sa

Appendix

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It is useful to know how the values of sin θ and cos θ for standard values of θ are derived, in addition to having memoπ rized them. First we begin with θ = = 45◦ : 4

x

π/4

h/2

x

h

h/2 π/4

 √ π , and the hypotenuse h is equal to x 2 + x 2 = x 2. 4 √ π x x 1 2 Therefore both the sine and cosine of are equal to = √ =√ = . 4 h 2 x 2 2

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

21

so we have

√ 3 2

√ 3 π = and cos = . x 6 2 √ √ x x 23 π π 3 Also, sin = = and cos = 2 = 3 x 2 3 x x 2

1 π = = sin 6 x 2

x

π/6 π/6

1 . 2

x

π/3

h

x/2

x

x/2 x

π/3

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π π = 30◦ and θ = = 60◦ : we take an equilateral triangle whose sides are all of length x, and all Next we look at θ = 6 3 π of whose angles are , and draw the perpendicular from the top vertex to the base, and in so doing bisecting the angle 3 at the top vertex.    √
2  2 3 3 x x 1 =x 1− =x =x The perpendicular bisector has length h = x 2 − = x2 − 2 4 4 4 2

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Review of Trigonometry for Calculus

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