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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]


Proving Identities Rigorously [01/23/2002]

Trigonometric Identities [11/13/1997]



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]

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]

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



Domain of Arccot [05/23/2002]

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]

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]

Functions of Complex Variables [02/20/2001]



Integration of a Trigonometric Function [06/10/1999]



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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