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 TOPICS This page:   trigonometry    Search   Dr. Math See also the Dr. Math FAQ:   trigonometry formulas See also the Internet Library:   trigonometry HIGH SCHOOL About Math Analysis Algebra    basic algebra    equations/graphs/      translations    linear algebra    linear equations    polynomials Calculus Complex Numbers Calculators/    Computers Definitions Discrete Math    permutations/    combinations Exponents    Logarithms Fibonacci Sequence/   Golden Ratio Fractals Functions Geometry    Euclidean/plane      conic sections/        circles      constructions      coordinate plane      triangles/polygons    higher-dimensional      polyhedra    non-Euclidean    practical geometry    symmetry/tessellations History/Biography Interest Logic Negative Numbers Number Theory Physics/Chemistry Probability Projects Puzzles Sequences/Series Sets Square/Cube Roots Statistics Transcendental   Numbers Trigonometry 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. Graphs and Gradients [05/13/2003] Sketch graphs of f and f' for f(x) = sin x and guess the formula for f'(x). 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 real height? 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 tank? 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 hill? 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 evaluate them? 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 considered infinite? 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 method. 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 the answer? 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 functions? 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 arccos respectively? 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? Page: []

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