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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.



Law of Cosines and Pythagorean Theorem [09/23/1997]
I would like to know how to calculate the angles in a triangle if all you know is the length of the three sides.

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.

Law of Tangents [01/22/1999]
What is the law of tangents? How is it derived?

Laws of Sines and Cosines [03/07/2002]
When I set up the problem, the three angles don't add up to 180 degrees.

Length of a Triangle's Sides [1/23/1995]
I have a triangle problem for you to solve: The lengths of the three sides of a triangle could be...

Length of the Diagonals of a Parallelogram [05/22/2000]
A parallelogram has a 70-degree angle and sides 6cm and 10cm long. How long are its diagonals?

Limit of Cos(Cos( ... Cos(x) ... ))) [02/24/2001]
Find the limit of a repeating trigonometric expression cos(cos( ... cos(x) ... )) as the number of cosines goes to infinity.

Location of Plane Flying on Great Circle [11/20/2003]
An aircraft flies from 60N 030W to 60N 030W following a great circle. When 025W is passed, what will the plane's latitude be?

Logarithmic Equivalent of the Inverse Hybolic Cosine Function [10/28/1997]
I recently attempted a question which I think I have done wrong and I was wondering if there were a set formula for this problem...

Maximizing Horizontal Distance [04/10/1998]
Using algebra, prove that an object needs to be thrown at 45 degrees to travel the greatest possible horizontal distance.

Mean Latitude/Longitude [07/10/2003]
Given three points on the earth measured in latitude/longitude, what is the formula to calculate a mean latitude/longitude for this group of 3?

Methods of Computing Trig Functions [09/07/99]
What would I do to figure out the actual angle of C, without using a table or calculator, if I know that tan(C) = 4/3?

Miter Angle of a Pipe [02/10/1999]
Find the miter angle of a pipe of certain shape.

Mitres on Pyramids [09/26/2002]
I am weatherproofing my home, and have to mitre boards in a pyramid with a rectangular - not square - base, and an apex that is directly over the centre of one edge of the base.

Mnemonic For Remembering Formulas for Sin, Cos, Tan [05/15/1998]
The story of Chief Soh Cah Toa.

Modeling Tides with Trigonometry [03/07/2001]
How can I find a trigonometric equation that models the depth of the water at the end of a pier throughout the day, given the heights and times of low and high tide?

Movement on a Sphere, Charted on a Spreadsheet [11/20/2010]
A student seeks help coding a spherical navigation spreadsheet program. Doctor Vogler helps him develop an algorithm that accounts for the trigonometry involved, with each drawing on archived conversations.

Moving a Circle on a Polar Graph [05/04/2001]
How do you move a circle so the center is not (0,0), but to where the center can be (r,[theta])?

Multiplying Large Numbers Using Sine and Cosine [12/17/2002]
I am researching Tycho Brahe and have come upon an example where he uses [sin(a + b) + sin(a - b)]/2 to multiply large numbers together due to the availability of sine tables. Can you explain the method?

Non-negative Acute Angle [01/26/1999]
How can you express the sine or cosine of any angle as the sine or cosine of a non-negative acute angle? Why would they be equal?

Octagon Formula [07/30/1997]
If you're building an octagon on a 12-foot radius, what is the length of each side?

Octagon Side Lengths [08/22/2001]
If I know that the dimension of an octagon from one side to the other is 8 feet, how can I find the lengths of a side?

One- and Two-sided Polygons [12/07/1999]
Can you explain what a monogon and a digon are?

Optimization: Minimum Area [11/07/1997]
How do you fold a piece of paper (rect. with width a and unlimited length) so one corner just reaches the righthand side for minimum area?

Ordering Products, Powers, and Parameters of Trigonometric Functions [10/31/2010]
A student wants to know how to unambiguously interpret strings of trigonometric functions, multiplication, and exponentiation. Doctor Peterson digs into a history book -- as well as another math doctor's conversation -- to illuminate the vagaries of the notation.

Origin of Radians [05/27/2002]
Where, exactly, did radians come from?

Origin of the Terms Sine and Cosine [07/02/1997]
I assume the term tan comes from the word tangent, but where do the terms sine and cosine come from?

Origin of the Terms Sine, Cosine, Tangent, etc. [10/27/1999]
Can you tell me the origin of the terms hypotenuse, sine, cosine, and tangent? Can you tell me how the trigonometric formulas for sine, cosine, and tangent came about?

Over the Right Field Fence [04/11/1998]
What would be the distance of a hit from home base over the top of a 30 ft. high right field fence?

Packing 4 Spheres Into a Tetrahedron [09/03/99]
How can I find the dimensions of the smallest tetrahedron that can serve as a container for 4 spheres packed as snugly as possible?

Pendulum Altitude [09/26/2002]
A pendulum 45cm long swings through a vertical angle of 30 degrees. Find the distance of the altitude through which the pendulum bob rises.

Perimeter of an Inscribed Regular Polygon [12/10/1998]
What is the formula for the perimeter of a regular polygon inscribed inside a circle?

Phase Difference [05/30/2003]
Does the sine wave lag or lead the cosine wave by pi/2?

Phase Shift in Sine Function [02/19/1999]
I am stuck on y = sin(4x+pi/3)... Could you please show me an example of a shifted graph?

Pilot's Rule of 60 [06/07/2002]
There is a famous piloting rule, called the Rule of 60, which you can use to steer clear of an obstacle. Why does it work? And is it possible to derive an inverse of the rule?

A Point Inside a Square [09/18/2003]
Point P lies inside square ABCD such that P's distance from A is 1, P's distance from B is 4 and P's distance from C is 5. What is the area of the square?

Polar Equations [05/05/2003]
If you have r^2 = 9*cos(theta) and r = 1 + cos(theta), how many points of intersection are there?

Precision in Measurement: Perfect Protractor? [10/16/2001]
Given that protractors are expected to be accurate to the degree, and in some instances the minute or second, how are angles accurately constructed and marked?

Premature Simplification [08/01/2003]
Solve: 2tan x + sin^2 x * sec x = 1 + sec x

Product Sum and Difference Formulae [11/20/1997]
(cos6theta+6cos4theta+15cos2theta+10)/(cos5theta+5cos3theta+10cos theta) = 2 cos theta.

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