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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.

Inverse Trig Functions and Special Angles [01/25/2006]
Find the exact value of the expression arctan(sqrt(3)). I know this means the angle with a tangent of sqrt(3) but how do I figure out that angle without a calculator?

Inverse Trigonometric Ratios [07/01/2003]
Why aren't the inverse trigonometric ratios equal to 1/(the ratio), as is the case for numbers?

Inverse Trigonometry [08/22/2001]
I can't find my mistake.

Irregular Sinusoidal Curves [08/24/2004]
Can you create a sinusoidal function with a fixed period such that the time span between the max and min is not the same as the time span between the min and max?

Land Plot Area [08/28/2002]
I have a parcel of land. The surveyor has determined it to be 624,630.88 square ft. and 14.340 acres. The legal description is 15.8 acres. How can I figure out the square footage?

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?

Meteoric Reprise [10/22/2016]
An adult sees a meteor, and wonders how close it came. With two applications of trigonometry laws, Doctor Rick compiles a table of distances.

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 How Much Wood Expands [09/21/2013]
A flooring inspector seeks to quantify how much wood buckles as it expands. Drawing on trigonometric approximations, Doctors Rick and Peterson each provide heuristics for this real world problem.

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?

Obtuse Angles Function [08/04/2014]
A student wonders why the relationship between trigonometric functions and triangle side ratios holds not just for right triangles, but for all triangles. Doctor Peterson models the trigonometry of a non-acute vertex by re-drawing it on the Cartesian coordinate system.

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?

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