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Selected answers to common questions:
Volume of a tank.
- 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
- Projectile Dynamics [10/21/2000]
A particle is projected with initial velocity v and angle theta in a
parabolic path. How can I show that at time t, when the angle to the
horizontal is gamma, tan(gamma) = tan(theta) - (gt/v) sec(theta)?
- Proof by First Principles [11/16/2001]
How can I prove by first principles that the derivative of tan(x) is
- Proof of a Trig Identity [03/13/1998]
Proving that (1 + sinX + cosX)/(1 + sinX - cosX) = (1 + cosX)/sinX.
- Proof of DeMoivre's Theorem [05/01/1997]
A typical induction proof: DeMoivre's theorem.
- Proof of the Parallelogram Law [08/07/1999]
How do you prove the parallelogram law geometrically, without using
- Proof of Trig Identity [7/24/1996]
Prove (sin 2t + sin t)/(cos 2t + cos t + 1) = tan t is an identity.
- Proving a Trigonometric Identity [06/06/1999]
How can I prove the trigonometric identities cos(a+b)*cos(a-b) = cos^2a -
sin^2b, and tan(x/2) = (1-cosx)/(sinx)?
- Proving a Trigonometry Identity [05/19/2003]
Prove the following identity: cos 3x = 4 cos^3x - 3cosx.
- Proving De Moivre's Theorem [12/03/1997]
Prove De Moivres theorem: - (cos(x)+isin(x))^n = cos(nx) + isin(nx) .
- Proving Identity for Sum of Cosines [10/17/2003]
How do you prove the identity for the sum of cosines?
- Proving Laws of Sines, Cosines [10/01/2003]
How do you prove the laws of sines and cosines?
- Proving Sine Identities [05/16/1998]
Can you help me prove the trigonometric identities for sin(A+B), sin(2A),
- Proving Trig Identities [11/15/1998]
How do I prove trig identities like sin^6x+cos^6x = 1-3sin^2xcos^2x?
- Proving Trig Identities vs. Solving Trig Equations [10/08/2006]
Proving the identity sinx/(1-cosx) - (1+cosx)/sinx = 0 is done a
little differently than solving that same equation for x, though the
thinking is very similar. What is the key to proving the identity?
- Proving Trigonometric Identities [02/12/1999]
Tips on proving trigonometric identities, as well as a good general
procedure to use, worked out on cos^4(x) - sin^4(x) = cos(2*x).
- Proving Trigonometry Identities [01/07/1997]
Prove these trigonometry identities (tan x+cot x)tan x = sec^2 x and
(1+cos x)(1-cos x) = sin^2 x
- Pythagorean and Other Triangle Theorems [1/6/1995]
Could you please show me the proof for the Pythagorean Theorem and list
the other triangle theorems that are based on it?
- The Pythagorean Identity for Cosine [11/11/1995]
If Sin 9 = x , what is the solution and result of Cos9.cos18.cos36?
- Pythagorean Identity of Trigonometry [02/16/1998]
Proof of the trigonometry identity sin^2 + cos^2 = 1.
- Radian Conversion [07/24/2012]
Used to performing trigonometry in degrees, a student gets thrown for a loop by
radians. Doctor Ian clarifies with an example based on an analogy of converting between
imperial and metric units of distance.
- Radians [03/22/2001]
I have searched everywhere for the history behind radians. Why did they
- Radian to Degree Conversions [11/11/2001]
Can you give me a table of radian measures and their reference degree
measures for the unit circle?
- Radius of Circumscribed Circle [05/11/2001]
Where can I find a derivation of R = abc/4K?
- Rail Bend in Hot Weather [10/13/2002]
A 20-ft piece of rail expands 1 in. in length during a hot spell. If
there are no expansion gaps, how high off the ground will the rail
- Raised Cosines [01/28/2001]
What are raised cosines, and how do they work?
- Ranges of Compositions of Inverse Trigonometric Functions [07/01/2017]
A young adult struggles to grasp the equivalence of the ranges of a pair of functions that compose sines, cosines, and their inverses. After suggesting a method, Doctor Peterson sketches the way forward.
- Ratios, Geometry, Trigonometry [06/10/1999]
A homeschool teacher asks for help with triangles, flagpoles, and
- Rearranging x=tan()sin() [05/01/2002]
How would I approach rearranging x = tan(y)*sin(y) into y = some
function of x?
- Reflections in Parabolic Mirrors [05/14/2005]
I know that in a parbola, any ray that starts at the focus and hits
the parabola is reflected parallel to the central axis of the
parabola. Can you explain or prove why that happens?
- Related Rates - Clock Hands [04/05/1999]
How fast is the distance between the tips of the hands on a watch
changing at one o'clock?
- Remembering Trig Functions [09/27/2001]
We have to memorize the sine, cosine, tangent, secant, cosecant, and
cotangent for 30, 45, and 60 degrees for precalculus class. Do you have
any tips on how to remember these?
- Replace Sin x with 1/n [12/19/2001]
Solve: (5 + 2sqrt(6))^(sin x) + (5 - 2sqrt(6))^(sin x) = 2sqrt(3).