2 Trigonometric Functions and Polar Co-ordinates
In this chapter we review trigonometry and periodic functions and discuss their relation to polar
co-ordinates. Some of this will be non-standard.
2.1 Definitions & Measuring Angles
Trigonometric functions date back at least 2000 years. Ancient mathemati-
cians were interested in the relationship between the chord of a circle and
the central angle, often for the purpose of astronomical measurement. It
wasn’t until 1595 that the term trigonometry (literally triangle measure) was
coined, and the functions were considered as coming from triangles.
Here are several related definitions of sine, cosine and tangent based either on triangles or circles.
Definition 2.1. 1. (a) Given a right triangle with hypotenuse (longest
side) 1 and angle θ, define sin θ and cos θ to be the side lengths
opposite and adjacent to θ.
Define tan θ =
sin θ
cos θ
to be the slope of the hypotenuse.
(b) Given a right triangle with angle θ, hypotenuse r, adjacent x and
opposite y, define
sin θ =
y
r
cos θ =
x
r
tan θ =
y
x
2. (a) ( cos θ, sin θ) are the co-ordinates of a point on the unit circle,
where θ is its polar angle measured counter-clockwise from the
positive x-axis. Provided cos θ = 0, also define tan θ =
sin θ
cos θ
.
(b) Repeat the definition for a circle of radius r with co-ordinates
(r cos θ, r sin θ).
Discuss some of the advantages and weaknesses of these definitions:
• What prerequisites are you assuming in each case?
• Is it easier to think about lengths rather than ratios?
• Where do you need basic facts from Euclidean geometry such as congruent/similar triangles?
• Convince yourself that that the triangle definitions follow from the circle definitions. What is
missing if you try to use the triangle definition to justify the circle version?
• If you were introducing trigonometry for the first time, what would you use?
If you’ve done sufficient calculus you might know of other definitions, for instance using power
(Maclaurin) series. Plainly these are not suitable for grade-school, but have the great benefit of
making the calculus relationship
d
dθ
sin θ = cos θ very simple. Establishing this using the triangle
definition is a somewhat tricky!
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