Glossary
Cosine
A trigonometric function that describes the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle, or the x-coordinate on the unit circle.
Example:
The cosine of 0 degrees is 1, representing the maximum horizontal displacement on the unit circle.
Degrees
A unit of angle measurement where a full circle is divided into 360 equal parts.
Example:
A right angle measures 90 degrees, which is π/2 radians.
Domain
The set of all possible input values (x-values) for which a function is defined and produces a real output.
Example:
The domain of the tangent function excludes angles where the cosine is zero, such as π/2, 3π/2, etc., because tangent would be undefined.
Domain Restrictions
Specific intervals or sets of values for which a function is defined, particularly important for inverse trigonometric functions to ensure they are one-to-one.
Example:
The domain restrictions for arcsin(x) mean that x must be between -1 and 1, inclusive, for the function to yield a real angle.
Inverse Trigonometric Functions
Functions that determine the angle corresponding to a given trigonometric ratio. They act as the 'undo' operation for sine, cosine, and tangent.
Example:
If you know the sine of an angle is 0.8, you use the inverse trigonometric function arcsin to find the angle itself.
Periodic Nature of Trig Functions
The property of trigonometric functions to repeat their values at regular intervals (periods), leading to infinite solutions for most trigonometric equations.
Example:
Because of the periodic nature of trig functions, if sin(x) = 0.5, then x = 30° is one solution, but so are 30° + 360°, 30° + 720°, and so on.
Radians
A unit of angle measurement where one radian is the angle subtended at the center of a circle by an arc equal in length to the radius.
Example:
A full circle measures 2π radians, which is equivalent to 360 degrees.
Range
The set of all possible output values (y-values) that a function can produce.
Example:
The range of the sine function is [-1, 1], meaning its output values will always be between -1 and 1, inclusive.
Sine
A trigonometric function that describes the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle, or the y-coordinate on the unit circle.
Example:
The sine of 90 degrees is 1, representing the maximum height on the unit circle.
Tangent
A trigonometric function that describes the ratio of the length of the opposite side to the length of the adjacent side in a right-angled triangle, or the ratio of sine to cosine.
Example:
The tangent of 45 degrees is 1, indicating that the opposite and adjacent sides are equal in length.
Trigonometric Equations
Equations that involve trigonometric functions of an unknown angle, requiring finding the specific angle(s) that satisfy the equation.
Example:
To find the angle where 2cos(x) + 1 = 0, you would solve this trigonometric equation for x.
Trigonometric Inequalities
Mathematical statements involving trigonometric functions and an inequality sign (e.g., >, <, ≥, ≤), requiring solutions over an interval.
Example:
Solving sin(x) > 0.7 involves finding all angles x for which the sine value is greater than 0.7, often visualized on a graph or unit circle.
Unit Circle
A circle with a radius of one unit centered at the origin of a coordinate plane, used to visualize trigonometric values and angles.
Example:
When solving cos(x) < -0.5, sketching the unit circle helps identify the quadrants where cosine is negative and less than -0.5.
arccos(x)
The inverse cosine function, which returns the angle whose cosine is x. Its range is typically restricted to [0, π] or [0°, 180°].
Example:
If cos(θ) = -1, then arccos(-1) gives θ = π radians or 180 degrees.
arcsin(x)
The inverse sine function, which returns the angle whose sine is x. Its range is typically restricted to [-π/2, π/2] or [-90°, 90°].
Example:
To find the angle whose sine is 1/2, you would calculate arcsin(1/2), which is π/6 radians or 30 degrees.
arctan(x)
The inverse tangent function, which returns the angle whose tangent is x. Its range is typically restricted to (-π/2, π/2) or (-90°, 90°).
Example:
When solving for an angle where tan(θ) = √3, you would use arctan(√3) to find θ = π/3 radians or 60 degrees.