Trigonometry is about ratios of right-triangle sides and acute angles {trigonometric function, mathematics}.
triangle sides
Right triangle has longest side opposite right angle {hypotenuse, right triangle}, side opposite acute angle {opposite side, right triangle}, and side adjacent to acute angle {adjacent side}.
ratios
Acute right-triangle angles have ratios of opposite side to hypotenuse {sine}, adjacent side to hypotenuse {cosine}, opposite side to adjacent side {tangent, angle}, adjacent side to opposite side {cotangent}, hypotenuse to opposite side {cosecant}, and hypotenuse to adjacent side {secant, trigonometry}.
sin = opposite/hypotenuse. cos = adjacent/hypotenuse. tan = opposite/adjacent. csc = hypotenuse/opposite. sec = hypotenuse/adjacent. cot = adjacent/opposite.
trigonometric relations
Tangent equals sine divided by cosine: tan = sin/cos. Cotangent equals cosine divided by sine: cot = cos/sin.
Sine equals cosecant reciprocal: sin = 1/csc. Cosine equals secant reciprocal: cos = 1/sec. Tangent equals cotangent reciprocal: tan = 1/cot. Cotangent equals tangent reciprocal: cot = 1/tan. Secant equals cosine reciprocal: sec = 1/cos. Cosecant equals sine reciprocal: csc = 1/sin.
domain and range
For all trigonometric functions, domain is all real numbers.
The sine and cosine range is from negative one to positive one. The secant range is from positive one to infinity. Cosecant range is from negative one to negative infinity. Tangent and cotangent range is all real numbers.
angles
Trigonometric functions can have angles of less than 0 or more than 90 degrees. Trigonometric functions can have angles between 270 and 360 degrees and negative acute angles. sin(A) = -sin(360 - A). tan(A) = -tan(360 - A). csc(A) = -csc(360 - A). cos(A) = cos(360 - A). cot(A) = -cot(360 - A). sec(A) = sec(360 - A). Trigonometric functions can have angles between 180 and 270 degrees and negative obtuse angles. sin(A) = -sin(A - 180). tan(A) = tan(A - 180). csc(A) = -csc(A - 180). cos(A) = -cos(A - 180). cot(A) = cot(A - 180). sec(A) = -sec(A - 180). Trigonometric functions can have obtuse angles between 90 and 180 degrees. sin(A) = sin(180 - A). tan(A) = tan(180 - A). csc(A) = csc(180 - A). cos(A) = -cos(180 - A). cot(A) = -cot(180 - A). sec(A) = -sec(180 - A).
angles: radians
Angle 360 degrees = 2*pi radians.
angles: phase
To make angle be from 0 to 360 degrees, add or subtract multiple of 360 degrees = 2 * pi radians. All trigonometric functions repeat values for angle plus 2 * n * pi and angle minus 2 * n * pi, where n is integer. For example, sin(A) = sin(A + 2 * n * pi) and sin(A) = sin(A - 2 * n * pi).
angles: differences
Trigonometric angle-difference functions relate to trigonometric angle functions. sin(A - B) = sin(A) * cos(B) - cos(A) * sin(B). cos(A - B) = cos(A) * cos(B) + sin(A) * sin(B).
angles: negative
Trigonometric negative-angle functions relate to trigonometric positive-angle functions. sin(-A) = -sin(A). cos(-A) = cos(A). tan(-A) = -tan(A).
angles: sums
Trigonometric angle-sum functions relate to trigonometric angle functions. sin(A + B) = sin(A) * cos(B) + cos(A) * sin(B), so sin(2*A) = 2 * sin(A) * cos(A). cos(A + B) = cos(A) * cos(B) - sin(A) * sin(B), so cos(2*A) = (cos(A))^2 - (sin(A))^2 = 2 * (cos(A))^2 - 1. tan(2*A) = (2 * tan(A)) / (1 - (tan(A))^2). tan(A) = (1 - cos(2*A)) / sin(2*A) = sin(2*A) / (1 + cos(2*A)).
sums and products
Sums of trigonometric functions relate to products of trigonometric functions. sin(A + B) + sin(A - B) = 2 * sin(A) * cos(B). sin(A + B) - sin(A - B) = 2 * cos(A) * sin(B). cos(A + B) + cos(A - B) = 2 * cos(A) * cos(B). cos(A - B) - cos(A + B) = 2 * sin(A) * sin(B). Set A = (x + y) / 2 and B = (x - y) / 2 to solve for sin(x) + sin(y), sin(x) - sin(y), cos(x) + cos(y), or cos(y) - cos(x).
Trigonometric functions relate to unit circle {circular function}.
Cosecant minus one is trigonometric function {excosecant} (excsc).
Secant minus one is trigonometric function {exsecant} (exsec).
Functions {hyperbolic function, trigonometry} can relate to unit hyperbola as trigonometric functions relate to unit circle. For independent variable x and base e of natural logarithms, (e^x - e^-x) / 2 {hyperbolic sine} (sinh). (e^x + e^-x) / 2 {hyperbolic cosine} (cosh). (e^x - e^-x) / (e^x + e^-x) {hyperbolic tangent} (tanh). (e^x + e^-x) / (e^x - e^-x) {hyperbolic cotangent} (coth). 2 / (e^x + e^-x) {hyperbolic secant} (sech). 2 / (e^x - e^-x) {hyperbolic cosecant} (csch).
relations
(sinh(x))^2 + (cosh(x))^2 = cosh(2*x). (cosh(x))^2 - (sinh(x))^2 = 1. sinh(x + y) = sinh(x) * cosh(y) + cosh(x) * sinh(y). cosh(x + y) = cosh(x) * cosh(y) + sinh(x) * sinh(y). arcsinh(x) = ln(x + (x^2 + 1)^0.5).
Trigonometric functions {principal branch function} can have domain 0 to 90 degrees, for acute angles.
If angle is acute, between 0 and 90 degrees, trigonometric functions have inverses {trigonometric inverse}: sin^-1 {arcsine}, cos^-1 {arccosine}, tan^-1 {arctangent}, cot^-1 {arccotangent}, sec^-1 {arcsecant}, and csc^-1 {arccosecant}.
One minus cosine is function {versed sine} {versine}. One minus the sine is function {coversed sine} {versed cosine} {coversine}.
Functions {wave function} can be waves.
series
Periodic functions can be trigonometric series. If period is T, series is: a0 + sum over i of (ai * cos(2n * pi * tau / T) + bi * sin(2n * pi * tau / T)), where a0 = (1/T) * (integral from -T/2 to T/2 of f(t) * dt), ai = (2/T) * (integral of f(t) * cos(2n * pi * t / T)), and bi = (2/T) * (integral of f(t) * sin(2n * pi * t / T)).
Sine or cosine can be zero. Even periodic function uses cosine. Odd periodic function uses sine.
period
For function over interval with width x, period T is twice interval length x: T = 2*x.
jump
Term coefficients depend on differences {jump} between left-hand and right-hand function limits, derivatives at jump points, and second derivatives at jump points.
analyzer
Harmonic analyzer can find first 20 coefficients from function graph and areas. Integrator circuits can calculate area.
Fourier trigonometric series {Fourier transform, function} can represent function over interval.
transformation
Function can transform by multiplying function by periodic function and integrating. Integral from x = -infinity to x = +infinity of g(x) * e^(-i * 2 * pi * u * x) * dx, where x = domain value, g(x) is function, u is frequency, and i is square root of -1.
transformation: coordinates
Fourier trigonometric series can transform coordinates. (1/(2 * pi)) * (integral from q = -infinity to q = +infinity of (e^(i*q*x)) * dq) * (integral from a = -infinity to a = +infinity of F(a) * (e^(-i*q*x)) * da).
limit
Fourier series go to limit as period approaches infinity.
time
Time t relates to phase A: t = tan(A/2).
power series
Complex power series can represent periodic functions with holomorphic positive and negative frequency.
Mechanisms {harmonic analyzer} can find first 20 Fourier-transform coefficients, using function graphs and measuring areas.
Functions {Fresnel integral} can have value C(z) = integral, from t = 0 to t = z, of cos(pi * t^2 / 2) * dt. It equals 0.5 if z equals infinity.
Functions {Gudermannian function} Gdx can have sin(u) = tanh(x), cos(u) = sech(x), or tan(u) = sinh(x) (Christoph Gudermann) [1798 to 1852].
One-half versed sine is a function {haversine} (hav).
Functions {Legendre function} S(z) can be integrals from t = 0 to t = z of sin((pi * t^2)/2) * dt and equal 0.5 if z equals infinity.
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Date Modified: 2022.0225