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Browse College Trigonometry
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to common questions:
    DeMoivre's theorem.



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/ AD)?

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 location.

Deriving Trigonometric Identities [07/20/2008]
How do you derive the sum-to-product and product-to-sum identities in trigonometry?

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 or right.

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?

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