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Selected answers to common questions:
Volume of a tank.
- 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?
- 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).
- Right Triangle Kite Problem [01/16/1997]
When flying a kite, 150 ft. makes an an angle of 51 degrees with the
ground. Assume the string is straight; how is the kite?
- Right Triangle Sides [09/14/1998]
To use SOHCOHTOA, I need help finding which sides are opposite and
- Roots of a Complex Equation [04/28/1997]
Given that cos t + isin t is the root of z^3 + az + b = 0 in which sin t
is not equal to 0, and that a and b are real and not equal to zero, show
that a + b^2 = 1.
- Rotating a Point around an Angle [05/16/1997]
How do you derive the formulas for rotating a point around an angle?
- Scaling a Right Triangle [12/01/2003]
Start with a right triangle of known dimensions. Move the hypotenuse
by some distance towards the right angle. What are the dimensions of
the new triangle?