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Trigonometry
50 flashcards
The half-angle formula for cosine is cos(x/2) = ±√((1+cos(x))/2)
The tangent and cotangent functions are reciprocals, so tan(x) = 1/cot(x) and cot(x) = 1/tan(x).
In a 45-45-90 triangle, sin(45°) = cos(45°) = √2/2 and tan(45°) = 1.
The law of sines relates the sides of any triangle to the sines of its opposite angles, and can be used to find unknown sides or angles.
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
The sine ratio of an angle in a right-angled triangle is the ratio of the length of the opposite side to the length of the hypotenuse.
The cosine ratio of an angle in a right-angled triangle is the ratio of the length of the adjacent side to the length of the hypotenuse.
The tangent ratio of an angle in a right-angled triangle is the ratio of the length of the opposite side to the length of the adjacent side.
The sine of an angle is the reciprocal of the cosecant of the same angle: sin(x) = 1/csc(x)
The fundamental trigonometric identity states that sin^2(x) + cos^2(x) = 1 for any angle x.
In a 30-60-90 triangle, sin(30°) = 1/2, cos(30°) = √3/2, tan(30°) = 1/√3, sin(60°) = √3/2, cos(60°) = 1/2, tan(60°) = √3.
The sum identity for cosine states that cos(x + y) = cos(x)cos(y) - sin(x)sin(y).
The double angle formula for sine is sin(2x) = 2sin(x)cos(x).
The condition sin(A)/a = sin(B)/b = sin(C)/c represents that the given triangle is similar to another triangle.
The area of a triangle with sides a and b, and included angle C is (1/2)absin(C).
The tangent ratio of an angle in a right triangle represents the slope of the line containing the opposite side.
The period of the sine function is 2π radians or 360 degrees.
The sum identity for sine states that sin(x + y) = sin(x)cos(y) + cos(x)sin(y).
The law of cosines relates the lengths of the sides of any triangle to the cosine of one of its angles, allowing finding unknown sides or angles.
The compound angle formula for cosine is cos(x + y) = cos(x)cos(y) - sin(x)sin(y).
For complementary angles x and (90° - x), sin(x) = cos(90° - x), cos(x) = sin(90° - x), and tan(x) = cot(90° - x).
The sum identity for tangent is tan(x + y) = (tan(x) + tan(y))/(1 - tan(x)tan(y)).
The secant ratio sec(x) = 1/cos(x) represents the ratio of the hypotenuse to the adjacent side in a right triangle.
The double angle formula for cosine is cos(2x) = 2cos^2(x) - 1 = cos^2(x) - sin^2(x).
The area of a triangle is proportional to its semi-perimeter times the product of the sines of its semi-angles (Heron's formula).
The reciprocal ratios are cosecant (csc) = 1/sin, secant (sec) = 1/cos, and cotangent (cot) = 1/tan.
The product-to-sum formula for tangent is tan(x)tan(y) = (tan(x+y))/(1-tan(x)tan(y)).
The sine law relates the sines of the angles of a triangle to the lengths of the sides opposite those angles.
The half-angle formula for sine is sin(x/2) = ±√((1-cos(x))/2)
The trigonometric functions of negative angles satisfy f(-x) = -f(x) for the sine and cosecant, and f(-x) = f(x) for the others.
The inverse of the sine function is called the arcsine or inverse sine function, denoted sin^-1(x) or arcsin(x).
The condition a^2 + b^2 = c^2 represents that the given triangle is a right triangle.
The theorem sin(A)/a = sin(B)/b relates two angles A and B in a triangle to the ratio of their opposite sides.
The law of tangents relates the lengths of two sides of a triangle to the tangents of half the difference of the opposite angles.
The tangent function has a period of π radians or 180 degrees.
Coterminal angles are angles that are in the same cycle on the unit circle and differ from each other by a multiple of 2π radians.
The tangent and cotangent functions have infinite limits as their input approaches π/2 or -π/2.
The side opposite the vertex angle A in a triangle is denoted as side a.
The double angle formula for tangent is tan(2x) = 2tan(x)/(1-tan^2(x)).
SOH CAH TOA is a mnemonic for recalling the definitions of sine, cosine, and tangent in right triangles: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
The triangle angle bisector theorem relates a triangle's sides to the opposite angles through the ratio (side a)/(side b) = sin(A)/sin(B).
The inverse of the cosine function is called the arccosine or inverse cosine function, denoted cos^-1(x) or arccos(x).
Reference angles are acute angles between 0 and 90 degrees that help evaluate trigonometric functions of any angle by relating it to angles in the first quadrant.
The trigonometric identity sin^2(x) + cos^2(x) = 1 represents the Pythagorean identity.
CAST represents the reciprocal trig ratios: Cosecant = 1/sin, Secant = 1/cos, Cotangent = 1/tan.
The area of a triangle with sides a and b, and included angle C is (1/2)absin(C).
The area of a triangle is proportional to the square of any side by the theorem: Area1/Area2 = (side1/side2)^2.
For complementary angles A and (90° - A), sin(A) = cos(90° - A), cos(A) = sin(90° - A), and tan(A) = cot(90° - A).
The tangent function is defined as the ratio of the sine and cosine functions: tan(x) = sin(x)/cos(x).
The range of values for the sine and cosine functions is [-1, 1].