Enhance Your Learning with Advanced Trigonometric Identities and Equations Flash Cards for quick learning
sin^2(x) + cos^2(x) = 1
csc(x) = 1/sin(x), sec(x) = 1/cos(x), cot(x) = 1/tan(x)
tan(x) = sin(x)/cos(x)
sin(π/2 - x) = cos(x), cos(π/2 - x) = sin(x), tan(π/2 - x) = cot(x)
sin(-x) = -sin(x), cos(-x) = cos(x), tan(-x) = -tan(x)
sin(x + y) = sin(x)cos(y) + cos(x)sin(y), cos(x + y) = cos(x)cos(y) - sin(x)sin(y)
sin(x - y) = sin(x)cos(y) - cos(x)sin(y), cos(x - y) = cos(x)cos(y) + sin(x)sin(y)
sin(2x) = 2sin(x)cos(x), cos(2x) = cos^2(x) - sin^2(x), tan(2x) = 2tan(x)/(1 - tan^2(x))
sin(x/2) = ±√((1 - cos(x))/2), cos(x/2) = ±√((1 + cos(x))/2), tan(x/2) = ±√((1 - cos(x))/(1 + cos(x)))
sin(x)sin(y) = (1/2)[cos(x - y) - cos(x + y)], cos(x)cos(y) = (1/2)[cos(x - y) + cos(x + y)], sin(x)cos(y) = (1/2)[sin(x + y) + sin(x - y)]
sin(x) + sin(y) = 2sin((x + y)/2)cos((x - y)/2), sin(x) - sin(y) = 2cos((x + y)/2)sin((x - y)/2), cos(x) + cos(y) = 2cos((x + y)/2)cos((x - y)/2), cos(x) - cos(y) = -2sin((x + y)/2)sin((x - y)/2)
Equations involving trigonometric functions, such as sin(x) = 0 or cos(x) = 1, which can be solved using algebraic techniques and trigonometric identities.
Functions that give the angle whose trigonometric value is a given number, such as arcsin(x), arccos(x), and arctan(x).
Visual representations of trigonometric functions, such as sine, cosine, and tangent, showing their periodic nature and key properties.
Demonstrations and derivations of various trigonometric identities using algebraic manipulations and geometric reasoning.
Real-world scenarios where trigonometry is used to solve problems, such as finding distances, angles, and heights in navigation, physics, and engineering.
In a triangle, the ratio of the length of a side to the sine of its opposite angle is constant: a/sin(A) = b/sin(B) = c/sin(C)
In a triangle, the square of a side is equal to the sum of the squares of the other two sides minus twice the product of their lengths and the cosine of their included angle: c^2 = a^2 + b^2 - 2abcos(C)
In a triangle, the ratio of the tangents of half the sum and half the difference of two angles is equal to the ratio of the lengths of the sides opposite those angles: (tan((A + B)/2))/(tan((A - B)/2)) = (a + b)/(a - b)
In a triangle, the ratio of the cotangents of half the sum and half the difference of two angles is equal to the ratio of the lengths of the sides adjacent to those angles: (cot((A + B)/2))/(cot((A - B)/2)) = (a + b)/(a - b)
e^(ix) = cos(x) + isin(x), where e is the base of the natural logarithm, i is the imaginary unit, and x is a real number.
(cos(x) + isin(x))^n = cos(nx) + isin(nx), where n is a positive integer and x is a real number.
Techniques used in calculus to simplify integrals involving algebraic expressions by substituting trigonometric functions.
The behavior of trigonometric functions as the input approaches certain values, such as infinity or zero, often used in calculus and analysis.
Infinite series involving trigonometric functions, used in the study of Fourier series and harmonic analysis.
Integrals involving trigonometric functions, often solved using trigonometric identities and techniques like substitution and integration by parts.
The derivatives of trigonometric functions, such as sin(x), cos(x), and tan(x), used in calculus and differential equations.
Inequalities involving trigonometric functions, often solved using algebraic techniques and trigonometric identities.
Approximations of trigonometric functions using Taylor series expansions or other mathematical techniques, useful in numerical analysis and engineering.
Identities involving complex numbers and trigonometric functions, used in complex analysis and the study of complex variables.
Equations involving trigonometric functions with multiple angles, such as sin(3x) or cos(2x), often solved using trigonometric identities and algebraic techniques.
Equations involving trigonometric functions and exponential functions, such as sin(x)e^x = 0, often solved using algebraic techniques and properties of exponential functions.
Equations involving trigonometric functions and logarithmic functions, such as sin(x)ln(x) = 0, often solved using algebraic techniques and properties of logarithmic functions.
Equations involving trigonometric functions and polynomial functions, such as sin(x^2) = 0, often solved using algebraic techniques and properties of polynomial functions.
Equations involving trigonometric functions and rational functions, such as sin(1/x) = 0, often solved using algebraic techniques and properties of rational functions.
Equations involving trigonometric functions and absolute value functions, such as |sin(x)| = 1, often solved using algebraic techniques and properties of absolute value functions.
Equations involving trigonometric functions and piecewise functions, such as f(x) = {sin(x), x < 0; cos(x), x ≥ 0}, often solved by considering different cases and applying algebraic techniques.
Equations involving trigonometric functions and parametric equations, such as x = sin(t), y = cos(t), often solved by eliminating the parameter and applying algebraic techniques.
Equations involving trigonometric functions and inverse trigonometric functions, such as sin(arcsin(x)) = 1, often solved by applying inverse trigonometric identities and algebraic techniques.
Equations involving trigonometric functions and hyperbolic functions, such as sin(x)cosh(x) = 0, often solved using algebraic techniques and properties of hyperbolic functions.
Equations involving trigonometric functions and special functions, such as Bessel functions or elliptic functions, often solved using specialized techniques and properties of the respective functions.
Equations involving trigonometric functions that cannot be solved analytically, often approximated using numerical methods like Newton's method or the bisection method.
Equations involving trigonometric functions that arise in the context of differential equations, often solved using techniques from differential equations and calculus.
Equations involving trigonometric functions that can be solved using power series expansions and techniques from calculus and analysis.
Equations involving trigonometric functions that can be solved using integral techniques, such as definite integrals or contour integrals.
Equations involving trigonometric functions that can be solved using matrix methods, such as eigenvalue analysis or matrix factorization techniques.
Equations involving trigonometric functions that can be solved using vector methods, such as vector algebra or vector calculus techniques.
Equations involving trigonometric functions that can be solved using techniques from complex analysis, such as residues or contour integration.
Equations involving trigonometric functions that can be solved using techniques from differential geometry, such as curvature or geodesic equations.
Equations involving trigonometric functions that can be solved using techniques from Fourier analysis, such as Fourier series or Fourier transforms.
Equations involving trigonometric functions that arise in the context of wave equations, often solved using techniques from partial differential equations and physics.