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Selected answers to common questions:
Volume of a tank.
- 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)?
- Deriving the Coordinates of the pi/6 Family [06/09/1999]
How do you derive the coordinates of the pi/6 family on the unit circle,
and why does this work?
- 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 the Law of Cosines [04/01/1998]
Will the Pythagorean Theorem work with a non-right triangle?
- Deriving Trig Functions; Taylor Series [05/01/2001]
How would I find, from first principles - no tables, no calculator - for
example, 32 degrees? If I use a formula, how is it derived?
- Deriving Trilinear Coordinates [05/18/1999]
How do you derive the trilinear coordinates of the orthocenter of a
- Determine Flagpole Height without Access to the Pole [12/01/2003]
I need to find the height of a flagpole, but am not allowed within 10
feet of the pole. I can't measure the 10 foot limit, which makes
accurate measurement of the pole's shadow impossible. How do I do it?
- Determining an Angle From Side Views [07/24/2002]
In the end elevation of a drawing, a rod rises from point A at 32
degrees. In the side elevation, the same rod is seen rising from point
A at 48 degrees. How do I work out at what angle to cut the end of the
- Determining Measure of Angle Without Tables or Calculators [03/05/1998]
Arctangent formula for determining measure of angle XOP, given P(x,y) on
the unit circle.
- Determining Triangle Similarity [05/26/1998]
Given two triangles, how can you determine if they are similar?
- Diameter of the Base of a Cone [08/12/1998]
How do you find the formula to calculate the diameter of the base of a
cone of nine degrees at various lengths?
- 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.
- Different Answers with Sine Rule and Cosine Rule [08/07/2002]
Triangle ABC has measurements AB=8.2cm, BC=9.4cm and AC=12.8cm and
angle A=47 degrees. Find angle B. Why does using the Sine Rule here
fail to give the right answer?
- 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?
- Dilation Designation: Why in the Family of Trigonometric Functions Rather than Quadratics? [09/08/2016]
An adult wonders why horizontal dilation falls under trigonometry rather
than, say, under quadratic function families: is there something inherent to
quadratics that makes such scale factoring redundant? Doctor Peterson algebraically isolates the multiplier, interprets what happens — and reveals the associativity at the heart of the matter.
- Distance from A to B [06/17/2001]
The bearing of B from C is 254 degrees. The bearing of A from C is 344
degrees. The bearing of A from B is 32 degrees. The distance from A to C
is 780 meters.
- Domain of Arccot [05/23/2002]
Why is the arccot function defined over 0 to pi, rather than from
-pi/2 to pi/2?
- Double- and Half-Angle Identities [11/24/1998]
Use the double- and half-angle trigonometry formulas to find the exact
values for sin(5 pi/12) and cos(7 pi/8).
- Double and Half-Angle Identities [12/08/2000]
Why do we have to learn the double and half-angle identities? Why can't
we just take the sine, cosine or tangent of double or half of the actual
- Drawing an Arrow [05/11/2000]
How can I find the coordinates of the edges of an arrowhead? I want a
program to draw an arrowhead from a user-defined baseline, with the angle
between the baseline and arrow side 30 degrees.
- Equations for Cartesian and Polar Coordinate Conversions [05/06/2003]
Given the following points in polar coordinates, find the rectangular
coordinates. (22, 51 degrees)
- Equations for the Paths of Four Lady Bugs [12/02/1998]
Four ladybugs travel from the corners of a square towards the center.
Find the equations that describe the path.
- Equilateral Triangle: Area Formula and Proof [06/16/1998]
Is there a formula to find the area of an equilateral triangle given a
point on its interior and the lengths of the segments from the point to
the three vertices?
- Etymologies of Algebra, Geometry, Trigonometry [10/22/2001]
What are the origins and roots of the words geometry, algebra, and
- Etymology of Cosine [5/21/1996]
Where does cosine come from?
- Euler Equation [01/21/1997]
What is the meaning behind e^(pi*i) = -1?
- Euler Equation [09/13/1997]
Does the Euler equation still work if we decide to work in degrees? Is it
- Euler's Theorem [05/09/2003]
Solve: sine(x) = 2.
- Evaluating a Trigonometric Expression [07/31/1999]
Can you help me evaluate tan (arcsin (3/5))?
- The Exact Value of the Sine of 1 Degree [05/24/2000]
Has anyone ever figured out the exact value for the sine of 1 degree or
- Explaining How SohCahToa Works [07/26/2004]
Can you please explain what "sohcahtoa" is all about so that I can understand it? I'd really appreciate it.
- Explaining the Dot Product [04/05/1998]
Exactly what does the dot product represent?
- Expressing Arccos in Terms of Arctan [02/24/1997]
How do you express Arccos in terms of Arctan?
- Extraneous Roots Introduced Trigonometrically [06/10/2012]
A student wonders what introduces extraneous roots into trigonometric identities and
problems. Doctor Peterson disentangles the root causes, and offers some remedies.
- Factoring: cos^3t + cos2t + sin^3 t = 0 [04/23/2003]
How can this expression equal 0? I need to find ALL solutions between 0 and pi.
- Factoring with Trigonometry [3/5/1996]
4tan^2X + 7tanX - 2 = 0
- Ferris Wheel [08/11/1997]
How can you find how high in the air you are when on a ferris wheel at
the 3 o'clock position?
- 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 ...
- Find Angle ACB [04/20/2002]
Let A', B' and C' be points on triangle ABC such that AA' BB' CC' are
angle bisectors. Suppose angle A'C'B' = 90 degrees. Find angle ACB.