MATH1013 Calculus I
Trigonometric ratios1
Edmund Y. M. Chiang
Department of MathematicsHong Kong University of Science & Technology
September 24, 2014
1Stewart, James, “Single Variable Calculus, Early Transcendentals”, 7th edition, Brooks/Coles, 2012
Briggs, Cochran and Gillett: Calculus for Scientists and Engineers: Early Transcendentals, Pearson 2013
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Trigonometric functions
Similar triangles
Some identities
Circular Functions
Inverse trigonometric functions
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Recalling trigonometric functions
• Trigonometric ratios have been known to us since Babylonians’time because of the need of astronomical observation. TheGreeks turns the subject into an exact science.
• Earliest definitions Given an angle A with 0 < A < π/2.Then one can form a right angle triangle 4ABC with the sidea opposite to the angle A called the opposite side, the sidenext to the angle A called the adjacent side, and the sideopposite to the right angle, the hypotenuse.
• One can certainly include the consideration that A = 0 orA = π/2 by accommodating a straight line section as ageneralized triangle, which will be useful when we discusstrigonometry ratios for arbitrary triangles later.
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Six trigonometric ratios
Figure: (Right angle triangle 4ABC : from Maxime Bocher, 1914)
sinA =opposite side
hypotenuse=
a
c, cosA =
adjacent side
hypotenuse=
b
c
tanA =opposite side
adjacent side=
a
b.
The other three are
cscA =1
sinA, secA =
1
cosA, cotA =
1
tanA
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Maxime Bocher’s book
Figure: (Bocher’s book (1914))
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Maxime Bocher’s bookMaxime Bocher (1867-1918) was an American mathematicianstudied in Gottingen, Germany, in late nineteenth century, then theworld centre of mathematics. He later became a professor inmathematics at the Harvard university.
Figure: (M. Bocher) Courtesy of Harvard University Archives
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Similar triangles
Figure: (Chapter One)
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Trigonometric Tables
Figure: (Chapter One)
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Right-angled triangle example I
Figure: Granville p. 3
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Right-angled triangle example II
Figure: Granville p. 4
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Some exercises
Figure: Granville p. 6
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Right-angled triangle π/4
Figure: Granville p. 4
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Right-angled triangle π/3 and π/6
Figure: Granville p. 5
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Special angle ratios
Figure: Granville p. 5
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
A special angle application
Figure: Granville p. 6
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Symmetry
• The point of the right-angled triangles show that triangleswith same angle share the same ratios.
• So any of the six ratios are invariant with respect to theactual dimensions (sizes) of the right-angled triangles.
• These ratios reflect some hidden secret of the threedimensional space we live in.
• As a matter of fact, these ratios appear from everyday lifeapplications to the most advanced scientific pursues.
• What we saw about the properties of trigonometric ratiosreflects our space has circular symmetry (i.e., symmetry to dowith circles).
• There are many more hidden symmetries in our space-time.
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
It really is about circles !!The crux of the matter of why any of the trigonometric ratios onlydepends on the angle and not on the size of the triangle reallymeans we are looking at a circle.
Figure: (Source: Wiki)
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Radian measures
Figure: 1.57, Briggs et al
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Central idea of radian measures
• The main idea is to use the ratio arc length of a circle to itsradius to represent the angle of the arc subtended at thecentre of the circle.
• The main point is that this ratio is independent of the radii.
• We start with full circles with radius r :
Full circle length
Radius=
2πr
r= 2π.
• The last expression illustrates that the ratio is independent ofthe r . That is, we normalize to unit circle
• For the general angle θ results from subtended from an arclength `r , we have
Part of full circle length
Radius=
Part of 2πr
r= Part of 2π = radian.
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Why radian measures?• Why we prefer radians over degrees?• Why 360◦ if we want a unit? 360 degrees is arbitrary• It is dimensionaless• The real advantage is the trigonometric functions are built in
for radian measure in Calculus:d
dθsin θ = cos θ,
d
dθcos θ = − sin θ
• But with degrees (or similarly with other non-radian units),
d
dθsin θ =
( π
180◦
)cos θ,
d
dθcos θ = −
( π
180◦
)sin θ
• The real reason is Euler formula:
e iθ = cos θ + i sin θ, e iπ = −1 (i2 = −1).
Otherwise, it would become
e iπθ/180◦
= cos θ + i sin θ.
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Trigonometric identities• The size of the right-angled triangle (circle) does not matter.
One immediately obtain the famous Pythagoras theorem
sin2 θ + cos2 θ = 1
where 0 ≤ θ ≤ π/2.• Dividing both sides by sin2 θ and cos2 θ yield, respectively
1 + cot2 θ = csc2 θ and tan2 θ + 1 = sec2 θ.
• Double-angle formulae:
sin 2θ = 2 cos θ sin θ, cos 2θ = cos2 θ − sin2 θ
• Half-angle formulae:
cos θ = 2 cos2θ
2− 1, sin θ = 1− 2 sin2 θ
2.
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Addition formulae
•
sin(x ± y) = sin x cos y ± cos x sin y
cos(x ± y) = cos x cos y ∓ sin x sin y .
•tan(x ± y) =
tan± tan y
1∓ tan x tan y
•
sin x cos y =1
2[sin(x + y) + sin(x − y)]
cos x cos y =1
2[cos(x + y) + cos(x − y)]
sin x sin y =1
2[cos(x − y)− cos(x + y)].
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Proof of Sine additional formula• We standardise the hypotenuse to unit length.
Figure: (Source: Wikipedia)
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Ratios beyond π2
• The question is now how to extend the angle beyond π2 , as
the unit circle model clearly indicates such possibility.
Figure: (Source: Bocher, page 34)
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Ratios as circular functions: 0 ≤ θ ≤ 2π• Due to the property of similar triangles, it is sufficient that we
consider unit circle in extending to 0 ≤ θ ≤ 2π with the ratiosaugmented with appropriate signs from the xy−coordinateaxes.
Figure: (Source: Bocher, page 38)
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Illustration of 0 ≤ θ ≤ π
Figure: Briggs, et al: 1.59
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Illustration of 0 ≤ θ ≤ 2π
Figure: Briggs, et al: 1.63
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Ratios beyond 2π
• For sine and cosine functions, if θ is an angle beyond 2π, thenθ = φ+ k 2π for some 0 ≤ φ ≤ 2π. Thus one can write downtheir meanings from the definitions
sin(θ) = sin(φ+k 2π) = sinφ, cos(θ) = cos(φ+k 2π) = cosφ
where 0 ≤ φ ≤ 2π, and k is any integer.
• The case for tangent ratio is slightly different, with the firstextension to −π
2 ≤ θ ≤π2 , and then to arbitrary θ. Thus one
can write down
tan(θ) = tan(φ+ k π) = tanφ
where θ = φ+ kπ for some k and −π2 ≤ φ ≤
π2 .
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Illustration of 2π ≤ θ ≤ 4π
Figure: Briggs, et al: 1.61
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Illustration of −4π ≤ θ ≤ −2π
Figure: 1.62 (Publisher)
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Periodic sine and cosine functions
Figure: (Source:Upper: sine, page 45; Lower: cosine, page 46)
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Periodic sine/cosec functions (publisher)
Figure: Briggs, et al: 1.63a
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Periodic cos/sec functions (publisher)
Figure: Briggs, et al: 1.63b
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Periodic tangent function
Figure: (Source: tangent, Borcher page 46)
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Periodic tangent functions (publisher)
Figure: 1.64a (publisher)
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Periodic cotangent functions (publisher)
Figure: Briggs, et al: 1.64b
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Graphs of −4π ≤ θ ≤ −2π
Figure: Briggs, et al: 1.62
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Inverse trigonometric functions• The sine and cosine functions map the [k 2π, (k + 1) 2π] onto
the range [−1, 1] for each integer k . So it is
many −→ one
so an inverse would be possible only if we suitably restrict thedomain of either sine and cosine functions. We note that eventhe image of [0, 2π] “covers” the [−1, 1] more than once.
• In fact, for the sine function, only the subset [π2 ,3π2 ] of [0, 2π]
would be mapped onto [−1, 1] exactly once. That is the sinefunction is one-one on [π2 ,
3π2 ].
• However, it’s more convenient to define the inverse sin−1 x on[−π
2 ,π2 ].
sin−1 x : [−1, 1] −→[− π
2,π
2
].
•cos−1 x : [−1, 1] −→
[0, π
].
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
One-one Sine
Figure: Briggs, et al: 1.71a
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Inverse sine
Figure: Briggs, et al: 1.72
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Inverse sine
Figure: Stewart: page 67
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
One-one Cosine
Figure: 1.71b (publisher)
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Inverse cosine
Figure: Briggs, et al: 1.73
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Inverse cosine
Figure: Stewart: page 68
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Inverse trigonometric functions II
• The tangent has
tan−1 x : (−∞, ∞) −→[− π
2,π
2
]or the whole real axis.
• Examples/Exercises• Find (a) sin−1(
√3/2), (b) cos−1(−
√3/2),
(c) cos−1(cos 3π), (d) sin(sin−1 1/2).• Given θ = sin−1(2/5). Find cos θ and tan θ, (p. 48)
cos−1(cos( 7π6 )).
• Find alternative form of cot(cos−1(x/4)) in terms of x .• Find cos(sin−1(x/3)).• Why sin−1 x + cos−1 x = π/2?
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Inverse tangent
Figure: Briggs, et al: 1.77
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Inverse cotangent
Figure: Briggs, et al: 1.78
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Inverse secant
Figure: Briggs, et al: 1.79
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Inverse cosec (publisher)
Figure: Briggs, et al: 1.80
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Remaining inverse functions
Figure: Stewart: page 69
Trigonometric functions Similar triangles Some identities Circular Functions Inverse trigonometric functions
Further examples
Find
• (a) tan−1(−1/√
3),
• (b) sec−1(−2),
• (c) sin(tan−1 x).