Glossary
CHOSHACAO
A mnemonic device used to remember the definitions of the reciprocal trigonometric functions (Cosecant, Secant, Cotangent) in terms of the sides of a right triangle (Hypotenuse, Opposite, Adjacent).
Example:
Using CHOSHACAO, you can quickly recall that Cotangent (CAO) is Adjacent over Opposite.
Cosecant (csc)
The reciprocal trigonometric function of sine, defined as 1/sin(x). In a right triangle, it is the ratio of the hypotenuse to the opposite side.
Example:
If a right triangle has an opposite side of 3 and a hypotenuse of 5, the sine of the angle is 3/5, and the cosecant of that angle is 5/3.
Cotangent (cot)
The reciprocal trigonometric function of tangent, defined as 1/tan(x) or cos(x)/sin(x). In a right triangle, it is the ratio of the adjacent side to the opposite side.
Example:
If a ramp has a slope (tangent) of 0.5, its cotangent would be 1/0.5 = 2, representing the ratio of horizontal distance to vertical rise.
Domain
The set of all possible input values (x-values) for which a function is defined. For reciprocal trigonometric functions, it excludes values where the denominator is zero, leading to vertical asymptotes.
Example:
The domain of sec(x) is all real numbers except odd multiples of π/2, as cosine is zero at these points.
Period
The smallest positive horizontal distance over which a function's graph repeats its pattern. For cosecant and secant, the period is 2π; for cotangent, it is π.
Example:
The graph of sec(x) completes one full cycle of its pattern every period of 2π radians.
Range
The set of all possible output values (y-values) that a function can produce. For cosecant and secant, the range excludes values between -1 and 1.
Example:
The range of csc(x) is (-∞, -1] U [1, ∞), meaning its output values are always greater than or equal to 1, or less than or equal to -1.
Reciprocal Functions
Functions that are the multiplicative inverse of another function. In trigonometry, they are formed by taking the reciprocal of the basic sine, cosine, and tangent functions.
Example:
If you know the value of sin(x), its reciprocal function cosecant(x) is simply 1 divided by that value.
Secant (sec)
The reciprocal trigonometric function of cosine, defined as 1/cos(x). In a right triangle, it is the ratio of the hypotenuse to the adjacent side.
Example:
When analyzing the path of a projectile, if the cosine of the launch angle is 0.8, the secant of that angle would be 1/0.8 = 1.25.
Trigonometric Equations
Equations that involve trigonometric functions of an unknown angle. Solving them often requires using identities and understanding function properties and domains.
Example:
Solving the trigonometric equation csc(x) = 2 involves finding the angles x for which sin(x) = 1/2.
Trigonometric Expressions
Algebraic expressions that contain trigonometric functions. These expressions can often be simplified using various trigonometric identities.
Example:
Simplifying the trigonometric expression (sec(x) * cos(x)) results in 1, because sec(x) and cos(x) are reciprocals.
Trigonometric Identities
Equations that are true for all values of the variables for which the expressions are defined. Reciprocal identities are fundamental examples.
Example:
The reciprocal trigonometric identity sec(x) = 1/cos(x) is always true for any x where cos(x) is not zero.
Unit Circle
A circle with a radius of one unit centered at the origin of a coordinate plane, used to define trigonometric functions for any angle based on the coordinates of points on its circumference.
Example:
On the unit circle, the cosecant of an angle is 1 divided by the y-coordinate of the point where the angle's terminal side intersects the circle.
Vertical Asymptotes
Vertical lines on a graph where the function approaches infinity or negative infinity, indicating values for which the function is undefined. For reciprocal trigonometric functions, they occur when the denominator (the primary trig function) is zero.
Example:
The function y = csc(x) has vertical asymptotes at x = nπ because sin(x) is zero at these points, making the function undefined.