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Selected answers to common questions:
Volume of a tank.
- 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?
- Secant, Cosecant, and Cotangent [01/16/1997]
How do you understand the secant, cosecant, and cotangent functions?
- Shadow Moving Along a Wall [08/16/1999]
There is a light at the front of a circular arena. A boy starting from B
runs at the rate of 3 m/s toward the center C. At what rate will his
shadow be moving along the rear wall when he is halfway from B to C?
- The Shortest Crease [12/29/1997]
A piece of paper is 6 units one side and 25 units on another side...
- Shot Out of a Cannon [10/12/1998]
A performer is shot out of a cannon at 30 degrees from the horizontal, to
land 20 meters away... What should his initial speed be?
- Signs of Sines (and Other Trigonometric Ratios) [07/11/2002]
The trigonometric ratios have the signs + or -, depending on the
quadrant. How do you know which signs to use in which quadrants?
- Simplifying Circle Formulas from the Dr. Math FAQ [04/30/2004]
In your FAQ on circle formulas, in the sections where the other five
values are derived from any two known values, could you write each
formula in terms of only the two known values, instead of using the
- Simplifying Expressions with Double-Angle Formulas [05/25/1998]
Using double-angle formulas, can you help me express 1 - 2*sin^2(4x) and
2*sin(5x)*cos(5x) as a single function?
- Simplifying Expressions with Trig Functions [04/04/2001]
How can I solve problems like: 4 csc(3pi/4) - cot(-pi/4)?
- Simplifying Trigonometric Equations [06/02/1998]
I need help simplyifying the following trigonmetric equation and proving
the trigonometric identity...
- Simplifying Trigonometric Expressions [9/13/1995]
A student asks for help simplifying and solving trigonometric
- Simplify Numerator and Denominator [4/15/1996]
What is the algebra, or actually the trig that lets you simplify: -
sin x + 1 / (1- sin x)^2
- Simulating Sound Waves [10/27/2000]
How can I simulate a sound wave that reaches a maximum height every
- A Sine, a Double Angle Sine, a Square Root -- and an Imaginative Substitution [05/19/2011]
A student of trigonometric identities seeks help with a proof involving sines of double and
quadruple angles, as well as a square root. Invoking de Moivre's Theorem to approach the
problem indirectly, Doctor Anthony manufactures a quadratic substitution with
particularly helpful root sums and products.
- Sine and Cosine Addition Formulas [11/29/2000]
Can you prove the addition formulas for sine and cosine? Also, if sin(2Q)
= a, then what is sin(Q) + cos(Q) in terms of a?
- Sine and Secant [04/30/2002]
Why isn't secant the reciprocal of sine and cosecant the reciprocal of
- Sine, Cosine, and Tangent on a Circle [04/14/1999]
Where are the sine, cosine, and tangent on a unit circle?
- Sine/Cosine Equations [10/23/1996]
Are there equations for sine or cosine?
- Sine, Cosine, Tangent [03/13/2001]
What are sine, cosine, and tangent?
- Sine, Cosine, Tangent; Four-meter Ladder [6/15/1996]
Is there any way to perform these functions without a calculator?
- Sine Function and Unit Circle [08/15/2003]
Solve for x, where x is an integer: sin(pi/x) = square root2/2.
- Sine numbers in Charts for Angles [12/22/1997]
I am a toolmaker and I'd like to know where the numbers in sine charts or
on a calculator come from.
- The Sine of 1 Degree [01/07/2001]
How can I find an exact value for the sine of 1 degree using only the
values of sine for special angles and the angle-addition formulas?
- Sine of 36 Degrees [11/18/2001]
Ptolemy calculated the sine of 36 degrees geometrically using the
construction of a regular pentagon. How did he do it?
- Sine of an Angle and Opposite Side [5/17/1996]
Given triangle ABC inscribed in a circle, prove why (sin A)/a equals the
diameter of the circle.
- Sines and equations [12/3/1994]
I am a tenth grade student taking AMF (advanced mathematical functions)
and was given the problem: 0<=X<=150; when does the Sin 8X = Sin X? I had
no problem doing this on my graphing calculator, and finding the
intersects between the functions, but is there any way to do this by
using some sort of equation?
- Sine Series [09/09/1998]
What does sin1*sin2*sin3***sin179 equal in degrees?
- Sine, Simply? [10/27/2011]
Taught to evaluate trigonometric expressions with a calculator, a student wonders if
there are simple formulas behind sine, cosine, and tangent. Doctor Jerry introduces the Maclaurin series and CORDIC algorithm.
- Sine Squared [06/27/2003]
What is sine squared, and how do you enter it in a calculator? How do
you use trig when there isn't a right triangle?
- Sin of More Than 2pi, 360 Degrees [8/6/1996]
How do you find the sin of 2020?
- sin(sqrt2) More Simply? [05/17/2017]
A teen seeks simpler forms for trigonometric functions of radicals. With examples,
Doctor Vogler explains how this pursuit leads to infinite cardinalities of sets and further number theory.
- Sin x^2, Sin^2 x, (Sinx)^2 [09/12/2001]
What is the difference between sin x^2, sin^2 x, and (sinx)^2?
- Sin(x) = x? [07/01/1999]
Is it true that when x is very small (e.g. x = 10^-70), then sin(x) = x?
- Solution for Trig Equality [7/27/1996]
Sin x = Cos x - what does it mean?
- Solve for x [10/26/2001]
Solve for x over (0,2pi): 1) tan^2 x = 2sec x - 1; 2) sin x - sqrt3 cos x
- Solving and Simplifying using Trig Identities [12/5/1995]
Could you please help? 1. Solve for x: 20.0^(3x) = 85.0 2. Simplify: 1 +
(sec^2(@) / cos^2(@)) 3. Show that tan^4(y) - sec^4(y) = 1 - 2sec^2(y)
- Solving an Equation with Sines and Exponents [05/24/2000]
How can I solve the equation sin^2(exp(c-x)) + sin^2(exp(c+x)) = 0, where
x is a (real) unknown and c is a real positive constant?