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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 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 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 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?

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 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 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 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; 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.

Find Height of a Flagpole without Approaching the Pole [04/13/2006]
Points A, B, and C are exactly on level ground, A to B is 80 meters, the angle at B is a right angle. On C is a vertical flagpole with the top of the pole = D. Angle CBD = 32 degrees and angle CAD = 24 degrees. What is the height of the flagpole CD?

Finding a Missing Angle [01/23/2002]
Using trigonometry, calculate the measure of angles ABC and ACB.

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