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Trigonometry Terms and Definitions

Angles & Measurement(17)
Functions(9)
Graphs(4)
Right Triangle(3)
AAdjacent SideRight TriangleAAmplitudeGraphsAAngleAngles & MeasurementAAngle in Standard PositionAngles & MeasurementAArc LengthAngles & MeasurementCCentral AngleAngles & MeasurementCComplementary AnglesAngles & MeasurementCCosecantFunctionsCCosineFunctionsCCotangentFunctionsCCoterminal AnglesAngles & MeasurementDDegreeAngles & MeasurementFFrequencyGraphsHHypotenuseRight TriangleIInitial SideAngles & MeasurementIInverse Trigonometric FunctionFunctionsNNegative AngleAngles & MeasurementOOpposite SideRight TrianglePPeriodGraphsPPeriodic FunctionFunctionsPPhase ShiftGraphsPPositive AngleAngles & MeasurementQQuadrantal AnglesAngles & MeasurementRRadianAngles & MeasurementRReference AngleAngles & MeasurementSSecantFunctionsSSectorAngles & MeasurementSSineFunctionsSSupplementary AnglesAngles & MeasurementTTangentFunctionsTTerminal SideAngles & MeasurementTTrigonometric RatioFunctionsUUnit CircleAngles & Measurement
33 of 33 terms
Angles & Measurement
AngleInitial SideTerminal SidePositive AngleNegative AngleDegreeRadianArc LengthCentral AngleUnit CircleSectorAngle in Standard PositionCoterminal AnglesQuadrantal AnglesReference AngleComplementary AnglesSupplementary Angles
Functions
SineCosineTangentCosecantSecantCotangentTrigonometric RatioPeriodic FunctionInverse Trigonometric Function
Graphs
AmplitudePeriodPhase ShiftFrequency
Right Triangle
HypotenuseAdjacent SideOpposite Side

33 terms

Angles & Measurement

(17 items)

Angle

A figure formed by two rays sharing a common endpoint (vertex), representing the amount and direction of rotation from one ray to the other.
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intuitionnotationcommon errorsrelated concepts
An angle measures rotation. One ray (the initial side) is fixed; the other (the terminal side) is obtained by rotating through a specified amount. The size of the rotation — not the lengths of the rays — determines the angle.
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Initial Side

The fixed ray from which an angle's rotation begins, lying along the positive xx-axis when the angle is in standard position.
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The starting line. Every angle begins here and rotates away from it. In the standard coordinate setup, the initial side always points to the right along the positive xx-axis.
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Terminal Side

The ray obtained by rotating the initial side through the given angle; its position determines all trigonometric function values.
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The ending line after rotation. Where the terminal side lands on the unit circle fixes the point (cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta) and therefore every trigonometric value of θ\theta.
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Positive Angle

An angle generated by counterclockwise rotation from the initial side to the terminal side.
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Counterclockwise is the positive direction by convention. A rotation of 90°90° counterclockwise is +90°+90°; the same rotation clockwise would be −90°-90°.
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Negative Angle

An angle generated by clockwise rotation from the initial side to the terminal side.
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Clockwise rotation produces a negative angle. The angles −90°-90° and 270°270° are coterminal — same terminal side, different rotation paths, different signs.
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Degree

A unit of angle measurement equal to 1360\frac{1}{360} of a full rotation, denoted by the symbol °°.
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A full turn is 360°360°, a half turn is 180°180°, a quarter turn is 90°90°. The number 360 persists because it is divisible by many small integers, making subdivision convenient.
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Radian

The angle subtended at the center of a circle by an arc whose length equals the radius: θ=sr\theta = \frac{s}{r}.
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A radian ties angle measurement directly to the circle. When the arc length matches the radius, the central angle is exactly one radian (≈57.3°\approx 57.3°). A full circle has 2π2\pi radians because the circumference is 2πr2\pi r.
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Arc Length

The distance along a circular arc intercepted by a central angle: s=rθs = r\theta, where θ\theta is in radians.
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Wrap a string around part of a circle and straighten it — that length is the arc length. The formula s=rθs = r\theta works cleanly because radians are defined as the ratio sr\frac{s}{r}.
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Central Angle

An angle whose vertex is at the center of a circle and whose sides are radii intercepting an arc.
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The angle seen from the center of the circle. It directly governs both the arc length (s=rθs = r\theta) and the sector area (A=12r2θA = \frac{1}{2}r^2\theta) of the region it carves out.
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Unit Circle

The circle of radius 11 centered at the origin, defined by x2+y2=1x^2 + y^2 = 1, whose points encode trigonometric values as coordinates.
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intuitionpropertiesrelated concepts
Setting the radius to 11 eliminates division from the trigonometric ratios. The coordinates of the point at angle θ\theta become the function values directly: x=cos⁡θx = \cos\theta, y=sin⁡θy = \sin\theta.
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Sector

The region enclosed by two radii and the arc between them, with area A=12r2θA = \frac{1}{2}r^2\theta where θ\theta is in radians.
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A pie-shaped slice of a circle. The sector's area is proportional to its central angle: half the circle's angle gives half the circle's area.
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Angle in Standard Position

An angle placed on the coordinate plane with its vertex at the origin and its initial side along the positive xx-axis.
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A universal reference frame for trigonometry. Fixing the vertex and initial side means every angle lands at a unique spot on the unit circle, making function evaluation systematic rather than case-by-case.
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Coterminal Angles

Two angles that share the same terminal side when placed in standard position, differing by an integer multiple of 360°360° (or 2π2\pi).
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Different amounts of rotation that land at the same place. The angles 50°50°, 410°410°, and −310°-310° all point the same direction — they produce identical trigonometric values.
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Quadrantal Angles

Angles whose terminal side lies along a coordinate axis: 0°0°, 90°90°, 180°180°, 270°270°, and their coterminal equivalents.
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These are the boundary angles between quadrants. The terminal side lands exactly on an axis, so one coordinate is 00 and the other is ±1\pm 1. This makes some functions undefined (division by zero) and others take their extreme values.
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Reference Angle

The acute angle between the terminal side of θ\theta and the xx-axis, always in [0°,90°][0°, 90°] (or [0,π2][0, \frac{\pi}{2}]).
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The reference angle strips away quadrant information and isolates the magnitude. Evaluate the function at the reference angle, then attach the sign dictated by the quadrant.
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Complementary Angles

Two angles whose measures sum to 90°90° (or π2\frac{\pi}{2} radians).
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In a right triangle, the two acute angles are always complementary. The word "co-" in cosine, cotangent, and cosecant comes from "complement" — each is the cofunction of its complement.
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Supplementary Angles

Two angles whose measures sum to 180°180° (or π\pi radians).
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Supplementary angles share the same sine but have opposite cosines: sin⁡θ=sin⁡(180°−θ)\sin\theta = \sin(180° - \theta) and cos⁡θ=−cos⁡(180°−θ)\cos\theta = -\cos(180° - \theta). This is because their terminal sides are reflections across the yy-axis on the unit circle.
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Functions

(9 items)

Sine

The trigonometric function that maps an angle θ\theta to the yy-coordinate of the corresponding point on the unit circle: sin⁡θ=y\sin\theta = y.
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propertiesnotationrelated concepts
Domain: all real numbers. Range: [−1,1][-1, 1]. Period: 2π2\pi. Odd function: sin⁡(−θ)=−sin⁡θ\sin(-\theta) = -\sin\theta. Zeros at θ=nπ\theta = n\pi.
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Cosine

The trigonometric function that maps an angle θ\theta to the xx-coordinate of the corresponding point on the unit circle: cos⁡θ=x\cos\theta = x.
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propertiesnotationrelated concepts
Domain: all real numbers. Range: [−1,1][-1, 1]. Period: 2π2\pi. Even function: cos⁡(−θ)=cos⁡θ\cos(-\theta) = \cos\theta. Zeros at θ=π2+nπ\theta = \frac{\pi}{2} + n\pi.
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Tangent

The ratio of sine to cosine: tan⁡θ=sin⁡θcos⁡θ=yx\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{y}{x}, geometrically the slope of the terminal side.
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propertiesnotationrelated concepts
Domain: all reals except π2+nπ\frac{\pi}{2} + n\pi. Range: (−∞,∞)(-\infty, \infty). Period: π\pi. Odd function. Strictly increasing on each period interval.
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Cosecant

The reciprocal of sine: csc⁡θ=1sin⁡θ\csc\theta = \frac{1}{\sin\theta}.
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Domain: all reals except nπn\pi. Range: (−∞,−1]∪[1,∞)(-\infty, -1] \cup [1, \infty). Period: 2π2\pi. Odd function. Undefined where sin⁡θ=0\sin\theta = 0.
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Secant

The reciprocal of cosine: sec⁡θ=1cos⁡θ\sec\theta = \frac{1}{\cos\theta}.
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propertiesnotationrelated concepts
Domain: all reals except π2+nπ\frac{\pi}{2} + n\pi. Range: (−∞,−1]∪[1,∞](-\infty, -1] \cup [1, \infty]. Period: 2π2\pi. Even function. Undefined where cos⁡θ=0\cos\theta = 0.
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Cotangent

The reciprocal of tangent, equivalently the ratio of cosine to sine: cot⁡θ=cos⁡θsin⁡θ\cot\theta = \frac{\cos\theta}{\sin\theta}.
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propertiesnotationrelated concepts
Domain: all reals except nπn\pi. Range: (−∞,∞)(-\infty, \infty). Period: π\pi. Odd function. Strictly decreasing on each period interval — the opposite of tangent.
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Trigonometric Ratio

A ratio of two sides of a right triangle relative to one of its acute angles, defining the six trigonometric functions geometrically.
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The central insight: these ratios depend only on the angle, not on the triangle's size. All right triangles sharing the same acute angle are similar, so their side ratios are identical. This makes the ratio a property of the angle alone.
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Periodic Function

A function ff for which there exists a positive constant TT such that f(x+T)=f(x)f(x + T) = f(x) for all xx in the domain. The smallest such TT is the fundamental period.
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intuitionpropertiesrelated concepts
The function repeats its values at regular intervals. Once you know the behavior over one period, you know it everywhere. All six trigonometric functions are periodic.
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Inverse Trigonometric Function

A function that reverses a trigonometric function on a restricted domain, returning the angle whose trigonometric value is the given input.
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intuitionpropertiescommon errorsrelated concepts
Trigonometric functions are periodic, so they fail the horizontal line test on their full domains. By restricting each to an interval where it is one-to-one (monotonic), an inverse can be defined that returns a unique angle.
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Right Triangle

(3 items)

Hypotenuse

The side of a right triangle opposite the right angle — always the longest side.
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The hypotenuse is fixed by the right angle, not by the acute angle under consideration. While "opposite" and "adjacent" swap depending on which acute angle is chosen, the hypotenuse stays the same.
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Adjacent Side

The leg of a right triangle that forms one ray of the acute angle under consideration (the other ray being the hypotenuse).
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"Adjacent" means next to. This leg touches the angle directly — it sits alongside the angle together with the hypotenuse. The adjacent side for one acute angle is the opposite side for the other.
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Opposite Side

The leg of a right triangle that lies directly across from the acute angle under consideration, not touching it.
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"Opposite" means across from. This leg does not touch the angle — it faces it from the far side of the triangle. Which side is "opposite" depends entirely on which acute angle is the reference.
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Graphs

(4 items)

Amplitude

The maximum vertical distance from the midline to a peak (or valley) of a sinusoidal function: for y=Asin⁡(Bx−C)+Dy = A\sin(Bx - C) + D, the amplitude is ∣A∣|A|.
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Amplitude controls the height of the wave. A larger ∣A∣|A| stretches the wave vertically; a smaller ∣A∣|A| compresses it. The function oscillates between D−∣A∣D - |A| and D+∣A∣D + |A|.
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Period

The horizontal length of one complete cycle of a periodic function: for y=Asin⁡(Bx−C)+Dy = A\sin(Bx - C) + D, the period is T=2π∣B∣T = \frac{2\pi}{|B|}.
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The period tells you how wide one wave is. A larger ∣B∣|B| compresses the wave horizontally (shorter period, faster oscillation); a smaller ∣B∣|B| stretches it (longer period, slower oscillation).
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Phase Shift

The horizontal displacement of a sinusoidal graph from its standard starting position: for y=Asin⁡(Bx−C)+Dy = A\sin(Bx - C) + D, the phase shift is CB\frac{C}{B}.
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Phase shift slides the wave left or right. Positive CB\frac{C}{B} shifts right; negative shifts left. The "standard" cycle begins at x=CBx = \frac{C}{B} instead of at x=0x = 0.
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Frequency

The number of complete cycles a periodic function completes per unit interval, equal to the reciprocal of the period: f=1T=∣B∣2πf = \frac{1}{T} = \frac{|B|}{2\pi}.
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Period measures how long one cycle takes; frequency measures how many cycles fit in a fixed interval. Higher frequency means faster oscillation and shorter period.
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