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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. Circle and Rectangle Area Problem [11/03/2003] One corner of a rectangle is on the center of a circle. The radius is larger than the small side of the rectangle but smaller than the large side. What is the area of their intersection? Circle of Radius 1 km around Given Latitude/Longitude [08/27/2002] I need feed points for a circle of radius 1 km with center at a given latitude and longitude. Eight or more points should be enough for the program. Circle on a Sphere Equation [05/05/2000] How can I find an equation for all points that fall within a given radius from a single latitude and longitude on the surface of a sphere? Circular Motion [01/27/2001] Two problems: An electric hoist is being used to lift a piece of equipment... A car is moving at a rate of 50 miles per hour and the diameter of its wheels is 2.5 feet... Circumference at a Given Latitude [01/26/2001] How can I calculate the circumference of a line around the earth (a circle) at 40 degrees north latitude? Circumference of a Globe [1/27/1996] How do you derive a formula to find the circumference of a large globe hung 25 feet in the air? Classifying Sides of a Right Triangle [06/12/2002] How do you find the adjacent side, opposite side, and hypotenuse of a triangle? Co-functions and Solving Trig Equations [11/18/2004] If sin(3x - 26)° = cos(5x - 60)°, find x. The Collapse of Compasses that Do Not Copy Segments, and the Lengths We Go To [04/16/2011] A teacher wonders, "Since when do collapsible compasses copy lengths?" Suggesting that Euclid may have posited this basic ability early among his propositions -- for the purposes of simplifying more interesting constructions -- Doctor Peterson then goes on to discuss the pedagogical pros and cons of compass quality, reliance, and over-reliance. Common Mistake in Simplifying Trig Expressions [12/07/2003] I don't know how to simplify this problem: (sin^2(x) - cos^2(x)) / (sin^2(x) - sin(x)cos(x)) Complex Numbers and Trigonometry [12/25/1996] If x is a real number, ArcSin(Sin(x)) = x. If z is a complex number, ArcSin(Sin(z)) does not equal z. Why? Complex Numbers, Trig Functions and Roots of 1 [10/30/1997] I'm convinced that, for an arc of length x in radians: (cos x + i sin x)^(2 pi / x) = 1... It's cool but why does it work? Computers and Trigonometric Functions [02/11/1997] How do computers and calculators calculate values for sine, cosine, and tangent? Computing Angles of a Right Triangle [09/04/1997] Is there a way to compute the other two angles of a right triangle without a calculator? If a table is required, where can I find it? Computing Sine and Cosine [8/27/1995] Is it possible to compute sine and cosine for a given angle without using a calculator or a sine table? Computing sin without a Chart [02/23/1998] Is there a formula for converting a sin angle back to degrees? Continued Fraction for Tan(x) [11/03/2000] Can you tell me why the "infinite continued fraction representation" for the tangent function works? Convergence of Product of Sines [10/17/2003] Prove that (sin(pi/n))*(sin(2pi/n))*...*(sin((n-1)pi/n)) = n/(2^(n-1)) for n >= 2. Cosine 20 Degrees [03/24/2002] What is the exact value of cosine 20 degrees? Cosine Addition Formula [12/13/1997] How can you prove the addition formula for cosine by using right triangles? Cosine Approximations [12/29/1997] You derived a cubic equation for cosine of 40 degrees - did you know this approximation...? Cosines and Sines [6/16/1996] Why does cos A = sin (90 - A)? Cosines of Three Angles [05/08/2003] Show that the largest angle of the triangle whose sides are 4, 5, and 6 is exactly twice the size of the smallest angle. Cotangent Contradiction [02/13/2015] Ratio of cosine over sine? or inverse of the tangent? An adult struggles to reconcile definitions of the cotangent function. After discovering the haphazard information returned by search engines for such a simple query, Doctor Peterson provides two foundational definitions. Covariants [05/02/1997] Express tan(pi/2 + X) as a single function of X or as a constant. Crossing a Canyon [5/10/1995] Basically, we're trying to cross a canyon. From a point on one side, a rope stretches across and drops ten feet vertically... Curious Composition a Matter of Degrees [02/18/2011] A student wonders about the curiously consistent output of a composition of trigonometric functions. Examining the range of the tangent, sine, and cosine functions, Doctor Peterson unravels the mystery. Cycloids [06/16/1998] What are cycloids, and how do they differ from sine curves? Defining the Arcsinh Function [08/09/2003] Find an exact value for x in: 4 sinh x = 3 Defining the Complement of an Angle [08/04/2007] If the complement of x is 90 - x, the complement is negative for x greater than 90. I've seen conflicting definitions of the complement, some saying it cannot be negative and others placing no restrictions. Which is correct? Definition of a Tangent [10/27/1995] What is the tangent of an angle? Degree Measure of a Central Angle [07/29/1997] Find the degree measure, to the nearest tenth, of a central angle whose intercepted arc measures 21 cm. in a circle of radius 4 cm. Degrees, Minutes, Seconds [08/26/1997] I know that degrees are what you use to measure angles, but I thought minutes and seconds were time measurements... DeMoivre's Formula [08/13/1998] Can you explain DeMoivre's Theorem? de Moivre's Formula [08/11/2002] How do I use de Moivre's formula to express cos(3p) and sin(3p) in terms of cos p and sin p? And why would this be useful? Demonstrating Sin, Cos, Tan on the Unit Circle [12/30/1998] Can you recommend a good way to demonstrate the trig functions sine, cosine, and tangent? What about on the unit circle? Derivation of Law of Sines and Cosines [11/02/1997] How do you derive the law of sines and the law of cosines? Derivation of Sum/Difference of Sine, Cosine, Tangent [02/16/2002] How can I find the derivation of the sum/difference of sine, cosine, and tangent? Derivations of Heron's Formula [11/24/1998] How is Heron's formula (Hero's formula) derived? Deriving a Trig Identity [04/09/1998] Can you derive the identity tan(x)+tan(y)+tan(z) = tan(x)tan(y)tan(z), where x, y, and z are angles of a triangle? Page: []

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