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Selected answers to common questions:
Volume of a tank.
- Circle of Radius 1 km around Given Latitude/Longitude [08/27/2002]
I need feed points for a circle of radius 1 km with center at a given
latitude and longitude. Eight or more points should be enough for the
- 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?
- Circular Motion [01/27/2001]
Two problems: An electric hoist is being used to lift a piece of
equipment... A car is moving at a rate of 50 miles per hour and the
diameter of its wheels is 2.5 feet...
- Circumference at a Given Latitude [01/26/2001]
How can I calculate the circumference of a line around the earth (a
circle) at 40 degrees north latitude?
- Circumference of a Globe [1/27/1996]
How do you derive a formula to find the circumference of a large globe
hung 25 feet in the air?
- Classifying Sides of a Right Triangle [06/12/2002]
How do you find the adjacent side, opposite side, and hypotenuse of a
- Co-functions and Solving Trig Equations [11/18/2004]
If sin(3x - 26)° = cos(5x - 60)°, find x.
- The Collapse of Compasses that Do Not Copy Segments, and the Lengths We Go To [04/16/2011]
A teacher wonders, "Since when do collapsible compasses copy lengths?" Suggesting
that Euclid may have posited this basic ability early among his propositions -- for the
purposes of simplifying more interesting constructions -- Doctor Peterson then goes
on to discuss the pedagogical pros and cons of compass quality, reliance, and over-reliance.
- Common Mistake in Simplifying Trig Expressions [12/07/2003]
I don't know how to simplify this problem: (sin^2(x) - cos^2(x)) /
(sin^2(x) - sin(x)cos(x))
- Complex Numbers and Trigonometry [12/25/1996]
If x is a real number, ArcSin(Sin(x)) = x. If z is a complex number,
ArcSin(Sin(z)) does not equal z. Why?
- Complex Numbers, Trig Functions and Roots of 1 [10/30/1997]
I'm convinced that, for an arc of length x in radians: (cos x + i sin
x)^(2 pi / x) = 1... It's cool but why does it work?
- Computers and Trigonometric Functions [02/11/1997]
How do computers and calculators calculate values for sine, cosine, and
- 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?
- Computing Sine and Cosine [8/27/1995]
Is it possible to compute sine and cosine for a given angle without using
a calculator or a sine table?
- Computing sin without a Chart [02/23/1998]
Is there a formula for converting a sin angle back to degrees?
- Continued Fraction for Tan(x) [11/03/2000]
Can you tell me why the "infinite continued fraction representation" for
the tangent function works?
- 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.
- Cosine 20 Degrees [03/24/2002]
What is the exact value of cosine 20 degrees?
- Cosine Addition Formula [12/13/1997]
How can you prove the addition formula for cosine by using right
- Cosine Approximations [12/29/1997]
You derived a cubic equation for cosine of 40 degrees - did you know this
- Cosines and Sines [6/16/1996]
Why does cos A = sin (90 - A)?
- Cosines of Three Angles [05/08/2003]
Show that the largest angle of the triangle whose sides are 4, 5, and
6 is exactly twice the size of the smallest angle.
- Covariants [05/02/1997]
Express tan(pi/2 + X) as a single function of X or as a constant.
- Crossing a Canyon [5/10/1995]
Basically, we're trying to cross a canyon. From a point on one side, a
rope stretches across and drops ten feet vertically...
- Curious Composition a Matter of Degrees [02/18/2011]
A student wonders about the curiously consistent output of a composition of trigonometric functions. Examining the range of the tangent, sine, and cosine functions, Doctor Peterson
unravels the mystery.
- Cycloids [06/16/1998]
What are cycloids, and how do they differ from sine curves?
- Defining the Arcsinh Function [08/09/2003]
Find an exact value for x in: 4 sinh x = 3
- Defining the Complement of an Angle [08/04/2007]
If the complement of x is 90 - x, the complement is negative for x
greater than 90. I've seen conflicting definitions of the complement,
some saying it cannot be negative and others placing no restrictions.
Which is correct?
- Definition of a Tangent [10/27/1995]
What is the tangent of an angle?
- Degree Measure of a Central Angle [07/29/1997]
Find the degree measure, to the nearest tenth, of a central angle whose
intercepted arc measures 21 cm. in a circle of radius 4 cm.
- Degrees, Minutes, Seconds [08/26/1997]
I know that degrees are what you use to measure angles, but I thought
minutes and seconds were time measurements...
- DeMoivre's Formula [08/13/1998]
Can you explain DeMoivre's Theorem?
- 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?
- Demonstrating Sin, Cos, Tan on the Unit Circle [12/30/1998]
Can you recommend a good way to demonstrate the trig functions sine,
cosine, and tangent? What about on the unit circle?
- Derivation of Law of Sines and Cosines [11/02/1997]
How do you derive the law of sines and the law of cosines?
- Derivation of Sum/Difference of Sine, Cosine, Tangent [02/16/2002]
How can I find the derivation of the sum/difference of sine, cosine, and
- Derivations of Heron's Formula [11/24/1998]
How is Heron's formula (Hero's formula) derived?
- Deriving a Trig Identity [04/09/1998]
Can you derive the identity tan(x)+tan(y)+tan(z) = tan(x)tan(y)tan(z),
where x, y, and z are angles of a triangle?
- Deriving Identities [08/26/1998]
I'm supposed to derive the identity for sin 3x in terms of sin x.
- Deriving the Arcsin Formula [01/24/1999]
Why is the arcsin(A) = pi/2 + bi, for some b? How can you derive the
inverse sine formula (i.e.: arcsin)?