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).
Mathematical Sciences>Algebra>Function>Kinds>Trigonometric
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Date Modified: 2022.0224