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Selected answers to common questions:
- Counting Extrema [10/23/1996]
Find the number of minima and maxima of the function [Sin Na/Sin a]^2 on
the interval [0, pi].
- Curious Property of a Regular Heptagon [04/06/2001]
How can I prove that in a regular heptagon ABCDEFG, (1/AB)=(1/AC)+(1/
- Proving Identities Rigorously [01/23/2002]
I've been taught that the proper way to work on identities is to work
with one side at a time only.
- Trigonometric Identities [11/13/1997]
Do you have any suggestions for finding an easy way to remember them that
I could apply to all identities?
- Algebraic Expression for cos(arctan(x)/3) [8/15/1996]
How do I find an algebraic expression for cos(arctan(x)/3) so that I can
get rid of the trigonometric operands?
- Amplitude of Function with Sine and Cosine [01/11/2004]
How do you predict the amplitude of a function involving both sine and
cosine? For instance, how is it possible to find the amplitude of
f(x) = a*sin(x) + b*cos(x) without using an automatic grapher?
- Arccos [08/23/1997]
Please solve: Arc cos (cos y) + Arc sin (sin x) = y + x.
- Asin/acos/atan for Complex Numbers [3/27/1996]
How do you find asin(x+iy), acos(x+iy), and atan(x+iy)?
- Bretschneider's Theorem and Cyclic Quadrilaterals [11/30/2000]
Can you prove Bretschneider's Theorem for the area of a quadrilateral?
Also, can you show that any quadrilateral with supplementary opposing
angles can be inscribed in a circle?
- Building a Circular Horse Pen [06/16/2002]
My Dad and I are building a round pen for our horse. We have 16
16ft. panels and a 10 ft. gate and a 4ft. gate. (270 ft. total) We
want to use a radius and mark the places to dig holes for each post
that will support the panels, but we don't know how long the radius
should be. Can you help?
- Catenary and Parabola Comparison [04/06/2004]
What is the difference between a catenary and a parabola? I don't
know the difference in shape. Why is the St. Louis arch a catenary
and not a parabola?
- Catenary Curve [03/30/1999]
Find the vertex of a catenary curve.
- A Change of Variables [05/05/2003]
How do I solve int(sin^5(2x)cos(2x)dx) ?
- Circle on a Sphere Equation [05/05/2000]
How can I find an equation for all points that fall within a given radius
from a single latitude and longitude on the surface of a sphere?
- Circles on the Surface of a Sphere [08/11/1999]
How can I convert the equation of a circle on a unit sphere in the
(X,Y,Z) coordinate system into a function that is defined in terms of
spherical coordinates theta and phi?
- Closed Form of Complex Function [03/24/2003]
I would like a closed form (not a power series) for f(z) such that f
is analytic and f(z) = 0 when z = (k*pi)^3, z = ((k*pi)^3)e^i2*pi/3,
and z = ((k*pi)^3)e^i4*pi/3 where k is a positive integer.
- Complex Variables [03/25/2003]
Is there any complex root for an equation like sin(x)=3/2?
What does a^i= ? where a is a real constant.
- Computing Angles of a Right Triangle [09/04/1997]
Is there a way to compute the other two angles of a right triangle
without a calculator? If a table is required, where can I find it?
- Convergence of Product of Sines [10/17/2003]
Prove that (sin(pi/n))*(sin(2pi/n))*...*(sin((n-1)pi/n)) = n/(2^(n-1))
for n >= 2.
- de Moivre's Formula [08/11/2002]
How do I use de Moivre's formula to express cos(3p) and sin(3p) in
terms of cos p and sin p? And why would this be useful?
- Derivation of Law of Sines and Cosines [11/02/1997]
How do you derive the law of sines and the law of cosines?
- Deriving Lagrange's Trig Identity [01/09/2004]
Using the identity 1 + z + z^2 + ... + z^n = (1 - z^(n+1))/(1 - z), z
not = 1, derive Lagrange's trig identity:
1 + cosx + cos(2x) + ... + cos(nx) = 1/2 + (sin[(2n+1)x/2])/(2sin(x/2))
where 0 < x < 2*pi.
- Deriving the Haversine Formula [04/21/1999]
I need to write a program module to calculate distances given longitude
and latitude data, trying to find an object within a mile's radius of its
- Deriving Trigonometric Identities [07/20/2008]
How do you derive the sum-to-product and product-to-sum identities in
- Difference Between arsinh and arcsinh Functions [05/13/2004]
Why isn't there a "c" in arsinh, arcosh, and artanh? These are the equivalents of arcsin, arccos and arctan but they are hyperbolic.
- Differing Definitions of arcsec(x) Lead to Confusion over Signs [12/03/2005]
I was trying to prove the derivative of arcsec(x), and according to my
book sec(arcsec x) equals |x|, but I think it should just be x. Can
you explain what I'm missing?
- Domain of Arccot [05/23/2002]
Why is the arccot function defined over 0 to pi, rather than from
-pi/2 to pi/2?
- Euler in the Product of a Regular Polygon's Diagonal Lengths [04/06/2010]
A professor emeritus considers an n-sided regular polygon A1, A2, ... An inscribed in
the unit circle; and conjectures that the product of the lengths of its diagonals equals
n. By defining the polynomial f(x) as the product of x - r over its (n - 1) roots, and
applying complex numbers and Euler's equation, Doctor Vogler proves that sin(pi/n) *
sin(2pi/n) * ... * sin[(n - 1)pi/n] = n/2^(n - 1).
- Explaining the Dot Product [04/05/1998]
Exactly what does the dot product represent?
- Find a Function, Integrals [12/17/1997]
Suppose the graph of f has the formula f(x)=-x+1 for 0<=x<=1; x-1
for 1<x<=2 ...
- Finding Arctan of a Complex Number [08/18/2008]
Can we separate the real and imaginary parts of arctan(x + iy)?
- Find the Cities [06/19/2001]
Given the latitudes and longitudes of Detroit, MI, and Miami, FL, find
every city within 10 miles of a straight line between them.
- Functions of Complex Variables [02/20/2001]
Do you know of a method of finding f(z) when u(x,y) is known, and f(z) is
analytic in a domain D? Also, how can I show that sin(x) + sin(2x) +
sin(3x) + ... + sin(nx) = 1/2 cot(x/2)cos[(n+1/2)x]/2sin(x/2)?
- Half-Angle Formula - Getting the Right Sign [06/11/2003]
Find the possible values of tan(1/2)theta when tantheta = 7/24.
- Inequality Involving Triangle Area and Sides [05/11/2003]
Prove that for any triangle with sides a, b, and c and area A, a^2+b^
2+c^2 is greater than or equal to 4sqrt(3)A.
- Integration of a Trigonometric Function [06/10/1999]
How to integrate the function arctan(sqrt(1-x^2)).
- Integration of Sin(x^2) [11/10/1997]
I have been given the solution in the form of Frensel's Sin, but it
explains nothing about how it was integrated. I am not looking for an
equation, I am looking for a reason!
- Inverse Hyperbolic Cosine in Terms of ln [05/27/2003]
What is t in the equation ln(e^0.1t + 9e^-0.1t) = ln10 ?
- Law of Cosines in Informal Trigonometric Proof [12/22/2004]
If cosA/cosB = b/a, then show that triangle ABC is either isosceles
- Limit of Perimeter/Area Ratio for Polygons [03/23/2000]
How can I show that as the number of sides of a polygon of given area
increases, the perimeter of the figure decreases and the limit of the
perimeter is the circumference of a circle of that given area?