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Selected answers to common questions:
Volume of a tank.
- Graphs and Gradients [05/13/2003]
Sketch graphs of f and f' for f(x) = sin x and guess the formula for
- Graphs of Sine and Cosine Functions [02/28/2003]
What is the relation between the sine and the cosine function? How is
the graph of y = cos(x) linked to the graph of y = sin(x)?
- Half-Angle Formula - Getting the Right Sign [06/11/2003]
Find the possible values of tan(1/2)theta when tantheta = 7/24.
- The Height of a Distant Tree [07/08/1998]
If a tree is 1 mile away, and I see it as 5 cm tall, how can I find its
- Height of a Hill [09/06/1997]
From the top of a hill, the angles of depression of two consecutive
kilometre stones due east are 30 and 45 degrees.
- Height of the Plane [03/29/2001]
Towers A and B are 4.1 miles apart on level ground. A pilot finds the
angles of depression to the towers to be 36.5 and 25 degrees,
respectively. Find the height of the airplane.
- Horizontal Gas Tank Content Formula [01/17/1999]
Can you help me find a formula to measure the volume of gas left in a
horizontal gas tank given the height of the gas and the dimensions of the
- How a Calculator Determines Values of Trig Functions [03/19/2005]
I'm interested in how my calculator finds the answer of sin(pi/4) or
tan(pi/6), for example. Does it use power series?
- How High Did My Rocket Go? [05/28/1999]
We were 15 feet away from a rocket at launch and measured an angle of 89
degrees on the first launch and 67 degrees on the second. Can you tell me
how high it went?
- How High is the Building? [03/08/2002]
Melissa and her friends decide to fly their paper airplanes out of the
window of a tall building...
- How Much Does a Beam Bulge When it Expands? [2/24/1996]
Imagine a railroad beam 1/4 mile long fastened at both ends. The beam
expands 2" one summer day, causing it to bulge up. How high does it go?
- How Tall is the Building on the Hill? [1/19/1995]
A building is 30 meters high and is on top of a hill. The angles of
elevation of the top and bottom of the building from a point at the foot
of the hill are 64 degrees and 58 degrees respectively. How high is the
- How the Trig Functions Got their Names [12/14/1997]
I can guess why three of the trig functions are called cosine, cotangent,
and cosecant. But why were the other three named the sine, the tangent,
and the secant?
- How to Derive cos 72 [6/19/1996]
Derive a radical expression for cos 72 using a 36-72-72 triangle.
- Hyperbolic Functions [06/27/1998]
What does the inverse of sinh or cosh mean?
- Hyperbolic Functions [10/14/1998]
Can you explain hyperbolic functions? How are they defined? How do you
- Inequality Involving Triangle Area and Sides [05/11/2003]
Prove that for any triangle with sides a, b, and c and area A, a^2+b^
2+c^2 is greater than or equal to 4sqrt(3)A.
- Infinity or Undefined? [06/21/1997]
Why is the tangent of 90 degrees considered undefined instead of being
- Inner Product [12/10/1997]
How can I find the angles of the triangle formed by three points using
the inner product?
- Inscribed Circle [3/18/1995]
I need the formula to find the radius of an inscribed circle in an
equilateral triangle. I also need to know the rationalization for the
- Integral of Trig. Function, Power [7/19/1996]
What is the integral of dx/cos^4 x?
- Integrals of the Cosecant, and of the Square Root of a Sum of Squares [03/05/2016]
A teen seeks general strategies for integrating a few trigonometric
expressions. Doctor Peterson obliges, providing the big picture as well
as troubleshooting a couple of her algebraic missteps — and revealing that her derivation is, in fact, equivalent to the answer in her textbook.
- Integrating a Product of Cosines [05/04/2005]
Let n and m be natural numbers. My book says that the integral from
-pi to pi of the function f(x) = cos(nx) * cos(mx) is pi if n = m and
0 else. I can't figure out how to prove this.
- Integrating a Quadruple-Angle Trigonometric Expression or Two [01/13/2011]
A network engineer seeks help integrating a trigonometric expression with a cosine
term in its denominator (or is it the cotangent?). Using double-angle identities and a
substitution, Doctor Ali provides a boost.
- Integration [5/29/1996]
What does integral of (cos(x).x(cub).sinh(x)square).dx mean, and what's
- Integration [02/17/1999]
Integrate x*tan^2 x with respect to x.
- Integration Hints [02/23/1999]
Can you help me integrate (cos[x])^4?
- Integration of Sin(x^2) [11/10/1997]
I have been given the solution in the form of Frensel's Sin, but it
explains nothing about how it was integrated. I am not looking for an
equation, I am looking for a reason!
- Integration of Trigonometric Functions [05/28/1999]
How can I integrate sin(x)sin(2x).dx ?
- Interpreting Expressions Involving Cosine Squared [03/16/2007]
I don't understand the difference between cos(x)^2, cos^2(x) and (cos
(x))^2. Are they all same?
- Inverse Functions [08/09/2002]
Please help with two problems: F(x) = 5x-6, G(x) = x-4: (f composed
with g)^-1; F(x) = 5x-6, G(x) = x-4: G^-1 composed with f^-1.
- Inverse Functions: Arcsec(x) [02/11/1999]
If y = arcsec(x) then what does x equal? How is this problem related to
- Inverse Hyperbolic Cosine in Terms of ln [05/27/2003]
What is t in the equation ln(e^0.1t + 9e^-0.1t) = ln10 ?
- Inverse of a Function vs. Inverse Proportionality [11/20/2002]
Finding the inverse of a function and graphing yields a graph that has
been reflected in the line y = x relative to the function. Inverse
proportionality, however, yields a reciprocal relation graphically.
Why do these two things have similar names yet mean different things?
- Inverses and Reciprocals of Functions [09/28/2005]
What is the difference between f^(-1)(x) and f(x)^(-1)?
- Inverse Sine of a Value Greater than One [12/22/2003]
If I know that sin(B) = 1.732, why can't I find angle B? When I try to
use my calculator it says, 'Error'. Does this have to do with the sine
curve or is it something else?
- Inverses of Trigonometric Functions [08/07/2002]
How can the functions cosine and sine have inverses of arcsin and
- Inverse Trig and Other Quadrants [11/05/2011]
For arcsin and other inverse trigonometric functions, how do you find solutions outside
the first quadrant? With diagrams and graphs, Doctor Peterson outlines a procedure.
- Inverse trig functions [12/14/1994]
I am trying to evaluate this: cot[sin^-1 (-1) + 2sec^-1 (-2)] In other
words... cotangent of inverse sine of negative one plus two times inverse
secant of negative 2.
- 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?