Trigonometry – Basic Proofs and Methods
Concepts of primary interest:
Geometric representations
Locus – the set of all point satisfying a condition
http://math2.org/math/algebra/conics.htm
Tools of the Trade
The unit circle
The conic sections are the intersections of plane with right circular cones as illustrated above. The
half angle of the cone is , and its complement is . The points common to a plane perpendicular to
the axis of the cones and one of the cones form a circle. If the plane is rotated such that its normal
makes an angle between 0 and  with respect to the cone axis, the locus of points common to the
plane and the cones is an ellipse. The locus of points common to the cones and a plane parallel to a
cone edge (normal makes an angle of  with respect to the cone axis) is the two sheets of a parabola.
If the plane is parallel to the cone axis, the locus of points common to the cones and the plane is the
two sheets of a hyperbola. There are degenerate cases that occur when the intersection includes the
vertex of the cone. As each figure is the intersection of a plane with cones, the figures are all plane
figures.
Contact: tank@alumni.rice.edu
The intersection of a cone and a plane parallel to
the circular base is a circle.
The intersection of a cone and a plane not parallel
to the circular base but intersects both sides of the
cone is an ellipse.
The intersection of a cone and a plane not parallel The intersection of a cone and a plane
to the circular base that intersects one side of the perpendicular to the circular base creates the
cone is a parabola.
branches of a hyperbola.
http://id.mind.net/~zona/mmts/miscellaneousMath/conicSections/circle.htm
The Circle:
Definition: A circle is the locus of points in a plane which are the same distance R, the radius, from
a common point, the center.
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Cartesian:
A circle of radius R centered at C = (a, b):
Polar:
A circle of radius R centered at (0,0):
Polar:
A circle of radius R centered at (d, ):
(x – a)2 + (y - b)2 = R2
r=R
r=
R2  c2  2cR cos(   )
The Ellipse:
Definition: An ellipse is the locus of points with the sum of their distances from two fixed points,
the focal points, is equal to a constant. ( r1 + r2 = 2 a.) The longest axis of the ellipse is its major
axis with length, twice the semi-major axis length a. The smallest distance across the ellipse is the
minor axis or twice the semi-minor axis distance b. The smallest distance between a focal point and
a point on the ellipse is called rp, the periapsis , and the smallest is ra, the apoapsis. (For planetary
orbits, the smallest distance from the (sun’s center) focus is the perihelion and the greatest is the
aphelion. For earth orbits, the distances are the apogee and perigee.) The difference between the near
and far distances is represented as ra – rp = e (2a) where e is the eccentricity of the ellipse, and 0  e
< 1. If the distances ra and rp are equal, the figure is a circle, an ellipse with eccentricity e zero.
Using r1 + r2 = 2 a, it follows that ra = (1 + e) a, rp = (1 - e) a and b = a 1  e2 . An integration using
the Cartesian representation yields expressions for the area of an ellipse: A = ab = a 1  e2 .
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Cartesian:
x2 y 2

1
a 2 b2
Polar: r 
p
(1  e2 ) a

1  e cos  1  e cos 
b = a [1 – e2]½
Distance from F1 to F2 is 2ea
(2e
a) 2
y2
y
F1
F2
2ea
The ellipse can also be defined as the locus of points whose distance from the focus is proportional
to the horizontal distance from a vertical line known as the conic section directrix, where the ratio is
< 1. That ratio is e, the eccentricity.
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http://mathworld.wolfram.com/Ellipse.html
Exercise: The semi-major axis length b can be found as an extremum of r sin. Using the equation
above, find an expression for b and find the angle  at which that extremum occurs.
Exercise: Use the Cartesian representation for an ellipse and integrate to find the area of an ellipse.
Exercise: Using r1 + r2 = 2 a and ra – rp = e (2a) , show that: ra = (1 + e) a, rp = (1 - e) a and
b = a 1  e2 . A figure will prove useful.
The Parabola:
As the eccentricity of an ellipse approaches 1, the figure elongates dramatically, and, for e = 1, the
curve opens into a parabola.
Definition: A parabola is the set of all point in a plane that are equidistant from a fixed line, the
directrix, and a fixed point, the focus.
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http://mathworld.wolfram.com/Parabola.html
Exercise:
Exercise:
Exercise: 
Hyperbola:
If the eccentricity of the conic section exceeds 1, the curves are hyperbolic.

Definition: A hyperbola is the locus of points for which the difference of their distances from two
fixed points, the focal points, is equal to a constant. ( r2 – r1 = 2 a.)
The hyperbola can also be defined as the locus of points whose distance from the focus is
proportional to the horizontal distance from a vertical line known as the conic section directrix,
where the ratio is > 1.
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http://mathworld.wolfram.com/Hyperbola.html
1  e2
Tools of the Trade
Curvature of a plane path: The radius of curvature of a path in two dimensions is the inverse of the
radius of the best-fit circle at each point along the path. A large circle has a small curvature, and a
small circle has alarge curvature. Recalling the definition of the radian measure of angles,
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the change in the direction of the path per distance s traveled
along the path is the curvature of the path (R-1 = /s.). The change
in the direction is measured in radians. In two dimensions, the
equation for the path is y(x). The direction  of the path can be
extracted from the expression for the slope of the path tan =
dy/ . The change in direction follows with some additional effort.
dx
If a small addition to a vector r is made in the direction of that 
vector, its magnitude changes while the direction of that vector is unchanged. A small addition
r
perpendicular to the original vector r changes its direction by  = 
r , but, to lowest order, not
its magnitude.
For the figure below,  is (dy) cos or cos dx (tan divided by
dx 2  dy 2 . 
It follows that:
 =
cos  dx  tan  )
dx  dy
2
2

cos   tan  )
1   tan  
2
 cos 2   tan  )
or (tan) = sec2 , a standard introductory calculus result.
R 1 

  tan  
 cos 2  

s
 x 
  tan 
 cos3  
 x
x 2  y 2
x
 d  dy  

3
  cos     

 dx  dx  
 
1  dy 2 
3
dx 

sec 

R 2

2
d y
d y
dx2
dx2
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Applying this result, the radius of curvature of an ellipse is a2b-1 at its intersections with the major
axis and b2a-1 at its intersections with the minor axis.
Problems
1.) Show that the distance from the center of an ellipse to one of its foci is c = a2  b2 and that the
eccentricity e = c/a.
2.) Show that the equation of an ellipse in polar coordinates with the center of the ellipse on the
origin is: r 2 =
a 2 b2
.
a 2 sin 2   b 2 cos 2 
3.) Start with the equation of an ellipse and show that the sum of the distances between a point on the
ellipse and the foci of the ellipse is equal to 2 a.

References:
1.
The Wolfram web site: mathworld.wolfram.com/
2.
http://math2.org/math/algebra/conics.htm
3.
http://id.mind.net/~zona/mmts/miscellaneousMath/conicSections/circle.htm
APPENDIX: Changing Vectors
Concepts of primary interest:
Vector attributes – magnitude and direction
Infinitesimal additions parallel – change magnitude
Infinitesimal additions perpendicular – change direction
A vector quantity has two attributes magnitude and direction. For small additions to a vector,
the claim is that a small piece added parallel to the original vector changes the magnitude of
the vector (not the direction) and that a small piece added perpendicular to the original vector
changes the direction (not the magnitude).
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In the drawing to the right, the original vector
v 
is v , and the small addition is v . The
v
original vector: v  v vˆ , the magnitude of the
v 
vector times its direction. The small addition
v
is represented in terms of its components
parallel and perpendicular to v .
v  v vˆ

v   v  vˆ   v  eˆ
v  v  vˆ  v  eˆ 
The direction parallel to v is its direction v̂ while the direction ê is perpendicular to v .
The drawing suggests that the magnitude of the vector increases by  v  and that the
direction of v  v makes an angle  of
 v 
v with respect to the original v . Recall that
the tangent is the side opposite over the adjacent and that  v  is small so that tan   .
These speculations are to be confirmed using more analytic techniques. A vector is its
magnitude times its direction ( v  v vˆ ) and the magnitude is the square root of the inner (dot)
product of the vector with itself ( v  v  v  v ). Rearranging: vˆ  v v  v
v v
.
The derivative process includes taking the limit that the changes become infinitesimal so its
machinery is to be invoked to handle the small part. First, the time rate of change of the
 magnitude of v is computed. (Note that v without the arrow is the magnitude always!)



dv d

dt dt

1v
v v  
2


dv
dt
 dv
v v
dt

v dv
 v  v  dv
v  dv
dt
dt
dt




v
v
v v



 dv
dt

Multiplying by t, it follows that:

dv dv

dt
 dt




dv
t  dv
dt
dt

t  v   v 
The change in the magnitude of the vector is just equal to the component of the small
addition that is parallel to the original vector. If the additional piece is anti-parallel then
 v  is negative and the magnitude is decreased.
Exercise: Identify the rules for differentiation used in the line:
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dv d

dt dt

1v
v v  
2


dv
 dv
dt
dt
v v

v dv
 v  v  dv
v  dv
dt
dt
dt




v
v
v v



 dv
dt
.
Attacking the direction with a time derivative yields:


dv
dv
dvˆ d  v
1  v 2 v  dt

dt
 


v  v 
dt dt 
v  v 2   v  v 3/ 2

   dv dt  v vˆ  v  dv dt    1


v
v
v dv dt   dv dt  vˆ
3
In the last pair of parentheses, the component of dv dt parallel to v is subtracted from dv dt
leaving only the perpendicular part of dv dt .
dv
dt

 dv
dt
 vˆ   dv dt 



dvˆ 1 dv
1

 dv
vˆ   dv
dt
dt
dt
dt v
v

eˆ





eˆ 

The last equation demonstrates that the component added perpendicular to the original vector
causes the change in the direction for the resultant as compared to the original. Multiplying
by t as before:
1
 vˆ   eˆ .
vˆ  dv  dv
vˆ  
v

dt

dt




v 

Exercise: The equation above seems to suggest that it is more difficult to change the
direction of a vector if it has a larger magnitude. Prepare a sketch that supports this
interpretation.
Problems
1.) The position vector for a particle in a plane can be represented as:
rP (t )  R(t ) cos( ˆ  sin( ) ˆj  where the angle  is a function of time. Compute
drP 
drP
d
 rP
dt

dt
dt . Resolve drP into components parallel to and perpendicular to rP . Compute
and compare it to  drP dt  . Compare rP1  drP dt  and d/dt.





 
2.) Given the initial vector v  400 iˆ  300 ˆj and the small addition v  4 iˆ  3 ˆj  3 iˆ  4 ˆj

compute the magnitudes of v and of v  v and the angles that each makes with respect to the x
axis. Use the equations developed in this handout to estimate v and v̂ . What does the quantity
vˆ
represent? Why was v divided into two parts? What are the magnitudes of the pieces
vˆ

 

enclosed in parentheses? Why is v  4 iˆ  3 ˆj  3 iˆ  4 ˆj resolved into the two terms in
parentheses?
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