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Trigonometry

Browse High School Trigonometry
Stars indicate particularly interesting answers or
good places to begin browsing.
Selected answers to common questions:
DeMoivre's theorem.
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
location.
 Deriving the Law of Cosines [04/01/1998]

Will the Pythagorean Theorem work with a nonright 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
triangle?
 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
rod?
 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 HalfAngle Identities [11/24/1998]

Use the double and halfangle trigonometry formulas to find the exact
values for sin(5 pi/12) and cos(7 pi/8).
 Double and HalfAngle Identities [12/08/2000]

Why do we have to learn the double and halfangle identities? Why can't
we just take the sine, cosine or tangent of double or half of the actual
angle?
 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 userdefined 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
trigonometry?
 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
arbitrary?
 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
pi/180 radians?
 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; x1
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.
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