Review of Trigonometry for Calculus 1 Iversit Un As

Review of Trigonometry for Calculus 1 iversit Un as

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S a s s si ka n Review of Trigonometry for Calculus tchewane 2002 Doug MacLean “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

o opposite h sin α = = , h hypotenuse o

a adjacent α cos α = = , and h hypotenuse a

o opposite tan α = = , a adjacent 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. 2 Review of Trigonometry for Calculus iversit Un as

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S a s s si ka n The Unit Circle: tchewane 2002 Doug MacLean In Calculus, most references to the trigonometric functions are based on the unit circle, x2 + y2 = 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.

1 radian

x (0,0) (1,0)

Figure 1. Review of Trigonometry for Calculus 3 iversit Un as

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S a s s si ka n Radian measure and degrees tchewane 2002 Doug MacLean Since the circumference of a circle is 2π times its radius, we have

2π radians = 360◦ = 4 right angles, so 360 ◦ 180 ◦ 4 right angles 2 1 radian = = = = right angles 2π π 2π π or 2π π 1 1◦ = 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 radian is, because The sine and cosine functions live on the unit circle! 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).Ifθ 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 4 Review of Trigonometry for Calculus iversit Un as

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S a s s si ka n tchewane 2002 Doug MacLean

θ sin θ

(0,0) cosθ

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

sin θ cos θ 1 1 tan θ = , cot θ = , sec θ = , csc θ = . cos θ sin θ cos θ sin θ Review of Trigonometry for Calculus 5 iversit Un as

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S a s s si Fundamental Angles of the First Quadrant: ka n tchewane 2002 Doug MacLean There are three acute angles for which the trigonometric function values are known and must be memorized by the student of Calculus. They are π π π (in radians) 6 , 4 , and 3 , (in degrees) 30◦,45◦, and 60◦, (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( ) π π π π radians 0 6 4 3 2

θ(degrees) 0 30 45 60 90

θ( ) 1 1 2 right angles 0 3 2 3 1 √ √ √ √ √ θ sin 2 2 2 2 2 √ √ √ √ √ θ cos 2 2 2 2 2 6 Review of Trigonometry for Calculus iversit Un as

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S a s s si Fill in the Blanks: ka n tchewane 2002 Doug MacLean

θ( ) π π π π radians 0 6 4 3 2

θ(degrees) 0 30 45 60 90

θ( ) 1 1 2 right angles 0 3 2 3 1 √ √ √ √ √ θ 0 1 2 3 4 sin 2 2 2 2 2 √ √ √ √ √ θ 4 3 2 1 0 cos 2 2 2 2 2

Simplify the Arithmetic:

θ( ) π π π π radians 0 6 4 3 2

θ(degrees) 0 30 45 60 90

θ( ) 1 1 2 right angles 0 3 2 3 1 √ √ θ 1 2 3 sin 0 2 2 2 1 √ √ θ 3 2 1 cos 1 2 2 2 0 Review of Trigonometry for Calculus 7 iversit Un as

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S a s Figure 3 shows these values on the ﬁrst quadrant of the unit circle. s si ka n tchewane 2002 Doug MacLean

y (0,1) — π ——13√ — ( , ——) 2 π 22 — √√22 3 ( —— , ) π 22— — 4 (√ —— 31 , ) π 22 — 6

0 (1,0) x

Figure 3. 8 Review of Trigonometry for Calculus iversit Un as

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S a s s si ka n Moving Beyond the First Quadrant tchewane 2002 Doug MacLean 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

−θ θ+π

Figure 4. Review of Trigonometry for Calculus 9 iversit Un as

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S a s s si ka n tchewane sin(π − θ) ≡ sin θ, cos(π − θ) ≡−cos θ 2002 Doug MacLean sin(θ + π) ≡−sin θ, cos(θ + π) ≡−cos θ sin(−θ) ≡−sin θ, cos(−θ) ≡ cos θ π π sin − θ ≡ cos θ, cos − θ ≡ sin θ 2 2

θ π π π π 2π 3π 5π π 7π 5π 4π 3π 5π 7π 11π 0 6 4 3 2 3 4 6 6 4 3 2 3 4 6 √ √ √ √ √ √ √ √ θ 1 2 3 3 2 1 − 1 − 2 − 3 − − 3 − 2 − 1 sin 0 2 2 2 1 2 2 2 0 2 2 2 1 2 2 2 √ √ √ √ √ √ √ √ θ 3 2 1 − 1 − 2 − 3 − − 3 − 2 − 1 1 2 3 cos 1 2 2 2 0 2 2 2 1 2 2 2 0 2 2 2 10 Review of Trigonometry for Calculus iversit Un as

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S a s Figure 5 is left blank for the student to ﬁll in: s si ka n (0,1) tchewane √— 2002 Doug MacLean (— , — ) π (— 13 , — ) — 22 2π 2 π √√—— (— , — ) — — ( — 22 , — ) 3 3 22 π π 3— — — (— , — ) 4 4 ( √ — 31 , — ) 22 π π 5— — 6 6

(-1,0) π 0 (1,0) x

π π 7— 11— 6 6 (—— , ) π π (—— , ) 5— 7— 4 4 4π 5 π (—— , ) — — ( —— , ) 3 3π 3 (—— , ) — ( —— , ) 2 Figure 5. (0,-1) Review of Trigonometry for Calculus 11 iversit Un as

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S a s s si π π ka n Periodicity All six trig functions have period 2 , and two of them, tan and cot have period : tchewane 2002 Doug MacLean 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) 12 Review of Trigonometry for Calculus iversit Un as

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S a s s si ka n Identities of the sine and cosine functions tchewane 2002 Doug MacLean 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 2 θ ≡ + 2 θ csc 1 cot 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: Review of Trigonometry for Calculus 13 iversit Un as

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S a s 2 2 2 s si c = a + b − ab γ ka n The Law of Cosines: 2 cos tchewane 2002 Doug MacLean

We have w = b sin(π − γ) = b sin γ, and z = b cos(π − γ) =−b cos γ,so c2 = w2 + (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 c2 slightly diﬀerently: z x = b α y = a β h = b α = a β π−γ cos , cos , sin sin , γ x2 = b2 − h2, y2 = a2 − h2,so w b a c2 = (x + y)2 = x2 + 2xy + y2 = b2 − h2 + (b α)(a β) + a2 − h2 = 2 cos cos c 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) so cos γ = sin α sin β − cos α cos β =−cos(π − γ) =−cos(α + β).

Therefore a b h cos(α + β) = cos α cos β − sin α sin β α xyβ c 14 Review of Trigonometry for Calculus iversit Un as

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S a s We collect the angle sum and diﬀerence formulae: s si ka n tchewane 2002 Doug MacLean

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

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

1 cos α cos β ≡ (cos(α + β) + cos(α − β)) 2 and if we subtract (3) from (4) we get

1 sin α sin β ≡ (cos(α − β) − cos(α + β)) 2 Review of Trigonometry for Calculus 15 iversit Un as

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S a s s si ka n Double Angle Formulae: tchewane 2002 Doug MacLean If we let β = α in (1) and (3) and divide by 2, we get:

sin 2α ≡ 2 sin α cos α (5) cos 2α ≡ cos2 α − sin2 α = 2 cos2 α − 1 = 1 − 2 sin2 α (6)

(6) leads to the two identities:

1 + cos 2α cos2 α ≡ (7) 2 1 − cos 2α sin2 α ≡ (8) 2 which in turn lead to the formulas 1 + cos 2α 1 − cos 2α cos α =± sin α =± . 2 2 16 Review of Trigonometry for Calculus iversit Un as

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S a s These in turn lead to the s si ka n tchewane Half-Angle Formulae: 2002 Doug MacLean

α 1 + cos α cos =± (9) 2 2 α 1 − cos α sin =± (10) 2 2

π π The above identities may be used to compute the exact values of trig functions at many other angles, such as = 1 π 8 2 4 and 12 , but in practice one usually uses a calculator or computer to get extremely accurate values of the trig functions. Review of Trigonometry for Calculus 17 iversit Un as

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S a s s si ka n Identities of the Other Four Trigonometric Functions tchewane 2002 Doug MacLean 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 cos α cos β tan α + tan β tan(α + β) ≡ ≡ ≡ ≡ , cos(α + β) cos α cos β − sin α sin β cos α cos β − sin α sin β 1 − tan α tan β cos α cos β π π sin(α + ) cos α −1 tan(α + ) ≡ 2 ≡ ≡ (thus the formula for slopes of perpendicluar lines). (α + π ) − α α 2 cos 2 sin tan sin(α + β) tan α + tan β tan(α − β) ≡ ≡ , cos(α + β) 1 + tan α tan β 2 tan α tan(2α) ≡ , 1 − tan2 α α 1 − cos α sin ± α 2 1 − cos α and tan = 2 = =± α + α 2 cos 1 + cos α 1 cos 2 ± 2 18 Review of Trigonometry for Calculus iversit Un as

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S a s s si ka n Inverse Trigonometric Functions tchewane 2002 Doug MacLean Deﬁnition: If h is a real number, the inverse sine or Arcsin of h is that number between −π/2 and π/2 whose sine is h. 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: If k is a real number, the inverse cosine or Arccos of k is that number between 0 and π whose cosine is k. 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. Review of Trigonometry for Calculus 19 iversit Un as

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S a s s si ka n Java Applets: tchewane 2002 Doug MacLean For an animated interactive look at these two functions, take a look at the applets ArcSine Applet and ArcCosine Applet

y y y=h x=h θ x θ x

θ=Arcsin h θ=Arcsin h

Figure 6. 20 Review of Trigonometry for Calculus iversit Un as

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S a s s si ka n Appendix tchewane 2002 Doug MacLean 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 memo- π rized them. First we begin with θ = = 45◦: 4

π/4h/2 h/2 π/4 h

π √ In a right angled isosceles triangle, the base angles are both equal to , and the hypotenuse h is equal to x2 + x2 = x 2. 4 √ π x x 1 2 Therefore both the sine and cosine of are equal to = √ = √ = . 4 h x 2 2 2 Review of Trigonometry for Calculus 21 iversit Un as

DEO PAT- π π ET RIÆ S ◦ ◦ a s s si θ = = θ = = x ka n Next we look at 30 and 60 : we take an equilateral triangle whose sides are all of length , and all tchewane 6 3 2002 Doug MacLean 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. √ x 2 x2 1 3 3 The perpendicular bisector has length h = x2 − = x2 − = x 1 − = x = x 2 4 4 4 2 so we have √ √ x 3 π 1 π x 3 sin = 2 = and cos = 2 = . 6 x 2 6 x 2 √ √ 3 x π x 3 π 1 π/6 π/6 Also, sin = 2 = and cos = 2 = . 3 x 2 3 x 2

xxh

π/3x/2 x/2 π/3 x