Graphs of sine and cosine

You will build the sine graph from a turning circle, then see how two numbers change its height and width. You will learn why amplitude measures height and why period measures the horizontal distance needed for a repeat. Next you will connect cofunctions and build the cosine graph, which has the same wave shape with a different starting place. Finally you will sketch complete cycles and read a graph to write its equation, using the instructor's examples and fresh practice problems.

Lessons

Vocabulary

Trigonometric function: Pronounced trig-uh-nuh-MET-rik FUNK-shun. A rule taking an angle or real-number turn as input and returning a ratio or circle-coordinate value. Like: A machine that accepts a turn and returns a number.
Sine: Pronounced syne. The height coordinate of a point on the unit circle; its graph begins at the middle and rises for small positive inputs. Like: Recording a wheel dot's height as the wheel turns.
Cosine: Pronounced KOH-syne. The sideways coordinate of a point on the unit circle; its graph begins at the top and falls for small positive inputs. Like: Recording how far right or left a wheel dot sits.
Input: Pronounced IN-put. The number you put into a function. On a function graph it is the horizontal coordinate. Like: The choice you enter into a vending machine.
Independent variable: Pronounced in-duh-PEN-dunt VAIR-ee-uh-bul. The variable you choose first, usually x; its value is the function's input. Like: The time you choose before checking the clock's hands.
Output: Pronounced OUT-put. The value returned by a function. On a function graph it is the vertical coordinate. Like: The item a machine gives back after your choice.
Dependent variable: Pronounced dih-PEN-dunt VAIR-ee-uh-bul. The variable determined by the input through the function, usually y. Like: A taxi fare that depends on the distance traveled.
xy-plane: Pronounced eks-why plane. The flat coordinate plane formed by a horizontal x-axis and a vertical y-axis. Like: A map with east-west and north-south directions.
Coordinate: Pronounced koh-OR-duh-nut. One number describing a point's position along an axis. Like: One part of a street address.
Ordered pair: Pronounced OR-derd pair. Two coordinates written as (x, y), horizontal first and vertical second. Like: Directions that say right first, then up.
Graph: Pronounced graf. A picture of the input-output pairs that satisfy a function's equation. Like: A height record drawn on moving paper.
x-axis: Pronounced eks AK-sis. The horizontal axis. Function inputs are read along it, and every point on it has y = 0. Like: The east-west line on a map.
y-axis: Pronounced why AK-sis. The vertical axis. Outputs are measured along it, and every point on it has x = 0. Like: The north-south line on a map.
Origin: Pronounced OR-ih-jin. The point (0, 0) where the axes meet. Like: The starting point of a map's directions.
Radian: Pronounced RAY-dee-un. An angle unit measuring signed rim distance divided by positive radius. Ordinary arc length divided by radius gives the angle's magnitude; π radians is 180°. Like: Measuring a turn by distance walked around a round track.
Real number: Pronounced REE-ul NUM-ber. Any point on the number line, including integers, fractions, and numbers such as π. Like: Any possible position along an endlessly long road.
Unit circle: Pronounced YOO-nit SUR-kul. The circle centered at the origin with radius 1; its point after angle θ is (cos θ, sin θ). Like: A round track always one unit from its center.
Arc length: Pronounced ark length. Distance measured along part of a circle's curved rim. Like: The length of crust on a pizza slice.
Central angle: Pronounced SEN-trul ANG-gul. An angle whose vertex is at the circle's center and whose sides meet the circle. Like: The corner at the tip of a pizza slice.
Domain: Pronounced doh-MAYN. All allowed inputs of a function. Like: All the choices a machine accepts.
Range: Pronounced raynj. All output values a function actually reaches. Like: All the heights a wheel dot can reach.
Continuous: Pronounced kun-TIN-yoo-us. A graph without gaps, breaks or jumps; small changes in input produce small changes in output. Like: A smooth road with no missing bridge.
Periodic function: Pronounced peer-ee-OD-ik FUNK-shun. A function with a positive repeat distance p such that f(x + p) = f(x) for every input. Like: A wallpaper pattern repeating along a wall.
Period (T): Pronounced PEER-ee-ud tee. The smallest positive horizontal repeat distance of a nonconstant sine or cosine wave. Like: The length of one repeat in wallpaper.
Cycle: Pronounced SY-kul. One complete run through a wave's repeating pattern; its horizontal length is the period. Like: One lap of a circular track.
Integer: Pronounced IN-tuh-jer. A whole-number count that can be positive, negative or zero; no fractional part. Like: Counting whole laps forward or backward.
Odd function: Pronounced od FUNK-shun. A function satisfying f(−x) = −f(x); changing the input sign changes the output sign. Like: Turning a picture halfway around its center.
Even function: Pronounced EE-vun FUNK-shun. A function satisfying f(−x) = f(x); opposite inputs give the same output. Like: A picture reflected across a vertical mirror.
Symmetry: Pronounced SIM-uh-tree. A flip or turn that makes a figure match itself. Like: Matching the two halves of a folded paper shape.
Symmetry about the origin: Pronounced SIM-uh-tree uh-BOUT thee OR-ih-jin. Every graph point (x, y) has a matching point (−x, −y). A half turn about (0, 0) preserves the graph. Like: A pattern unchanged when its paper is turned halfway around.
Symmetry about the y-axis: Pronounced SIM-uh-tree uh-BOUT thee why AK-sis. Every point (x, y) has a matching point (−x, y), so the left and right halves mirror each other. Like: Two sides of a face seen around its center line.
With respect to: Pronounced with ree-SPEKT too. Relative to the named object or viewpoint; abbreviated w.r.t. in the notes. Like: Giving directions from a particular landmark.
Constant: Pronounced KON-stunt. A quantity that stays fixed while the input changes. Like: A fixed camera setting while a wheel turns.
A: Pronounced ay. The outside coefficient multiplying sine or cosine. Its size changes height and its sign can reverse the wave. Like: The enlargement setting on a copier.
ω (omega): Pronounced oh-MAY-guh. The inside coefficient multiplying x. Its size controls how quickly the angle changes and therefore how wide a cycle is. Like: The speed setting on a turning wheel.
Absolute value: Pronounced AB-suh-loot VAL-yoo. A number's distance from 0; it keeps size and removes a negative sign. Like: Distance from home without east or west direction.
Amplitude: Pronounced AM-plih-tood. The vertical distance from a wave's midline to its top or bottom; half the maximum-to-minimum distance. Like: How far a wheel rim reaches above its center.
Midline: Pronounced MID-lyne. The horizontal line halfway between a wave's maximum and minimum. Like: The center height of a wheel.
Maximum: Pronounced MAK-sih-mum. The largest output reached by a function in the set being considered. Like: The highest floor an elevator reaches.
Minimum: Pronounced MIN-ih-mum. The smallest output reached by a function in the set being considered. Like: The lowest floor an elevator reaches.
Local maximum: Pronounced LOH-kul MAK-sih-mum. A peak whose output is at least as high as nearby outputs. These sine and cosine peaks also reach the whole wave's maximum. Like: A hilltop compared with the ground nearby.
Local minimum: Pronounced LOH-kul MIN-ih-mum. A trough whose output is at least as low as nearby outputs. These sine and cosine troughs also reach the whole wave's minimum. Like: A valley floor compared with the ground nearby.
Vertical stretch or compression: Pronounced VER-tih-kul strech or kum-PRESH-un. Multiplying all outputs by A changes their distances from the midline by the factor |A|. Like: Resizing a photo's height while keeping its width.
Vertical stretch: Pronounced VER-tih-kul strech. Multiplying outputs by a coefficient of size greater than 1 moves points farther from the midline. Like: Making a photo taller.
Vertical compression: Pronounced VER-tih-kul kum-PRESH-un. Multiplying outputs by a nonzero coefficient of size less than 1 moves points closer to the midline. Like: Making a photo shorter.
Horizontal stretch or compression: Pronounced hor-ih-ZON-tul strech or kum-PRESH-un. Changing |ω| changes each wave's width: larger |ω| gives a shorter period, and smaller nonzero |ω| gives a longer period. Like: Resizing a photo's width while keeping its height.
Horizontal stretch: Pronounced hor-ih-ZON-tul strech. An inside coefficient with 0 < |ω| < 1 lengthens the period beyond 2π. Like: Pulling a repeating pattern wider.
Horizontal compression: Pronounced hor-ih-ZON-tul kum-PRESH-un. An inside coefficient with |ω| > 1 shortens the period below 2π. Like: Squeezing a repeating pattern into less space.
Reflection: Pronounced ree-FLEK-shun. A mirror flip of a graph. Negative A flips vertically across the x-axis; negative ω flips horizontally across the y-axis. Like: Seeing a picture in a mirror.
Point plotting: Pronounced point PLOT-ing. Finding input-output pairs, placing their points, and using the function's shape to connect them. Like: Marking landmarks before drawing a route.
Key points: Pronounced kee points. Five points spaced one quarter period apart that anchor one sine or cosine cycle. Like: Five landmarks around a complete route.
Intercept: Pronounced IN-ter-sept. A point where a graph meets an axis. An x-intercept has y = 0; a y-intercept has x = 0. Like: A route crossing a main road.
Quarter period: Pronounced KWOR-ter PEER-ee-ud. One fourth of the full cycle's horizontal length, T ÷ 4. Like: One of four equal stretches around a lap.
Cofunctions: Pronounced KOH-funk-shunz. The partner pairs sine/cosine, tangent/cotangent, and secant/cosecant; each matches its partner at the quarter-turn-minus-angle argument where defined. Like: Two viewpoints that exchange horizontal and vertical jobs.
Cofunction identities: Pronounced KOH-funk-shun eye-DEN-tih-teez. Equalities replacing a trig function by its cofunction at $π over 2$ − x, or 90° − x, wherever both sides are defined. Like: The same measurement described from the other corner.
Complementary angles: Pronounced kom-pluh-MEN-tuh-ree ANG-gulz. Two angles whose measures add to 90°, or $π over 2$ radians. A cofunction argument can be negative beyond the acute-angle picture. Like: Two pieces filling one square corner.
Tangent: Pronounced TAN-junt. Sine divided by cosine where cosine is not zero; abbreviated tan. Like: The rise of a ramp divided by its horizontal run.
Cotangent: Pronounced koh-TAN-junt. Cosine divided by sine where sine is not zero; abbreviated cot and paired with tangent as a cofunction. Like: A ramp's horizontal run divided by its rise.
Secant: Pronounced SEE-kant. One divided by cosine wherever cosine is not zero; abbreviated sec and paired with cosecant as a cofunction. Like: Ladder length per unit of floor distance.
Cosecant: Pronounced koh-SEE-kant. One divided by sine wherever sine is not zero; abbreviated csc and paired with secant as a cofunction. Like: Ladder length per unit of height reached.
Identity: Pronounced eye-DEN-tih-tee. An equation true for every input where its two sides are defined. Like: Two different packages holding the same amount.
Undefined: Pronounced un-dih-FYND. Having no assigned value, such as when a function's formula would require division by zero. Like: A machine with no valid choice for that request.
Sinusoidal function: Pronounced sy-nuh-SOY-dul FUNK-shun. A function whose graph is a sine-shaped or cosine-shaped wave, including its stretches, reflections and shifts. Like: A repeating smooth up-and-down record of a turning wheel.
Horizontal shift: Pronounced hor-ih-ZON-tul shift. Sliding a graph left or right without changing its shape or height. Like: Moving wallpaper sideways without stretching it.
Exact value: Pronounced ig-ZAKT VAL-yoo. A value written without rounding, often as a fraction, π expression or square root. Like: Saying one third instead of stopping at 0.33.
Rounded decimal: Pronounced ROWN-did DES-ih-mul. A nearby decimal keeping a chosen number of places; it may differ from the exact value. Like: Measuring to the nearest mark on a ruler.
Coefficient: Pronounced koh-uh-FISH-unt. A number multiplying a variable or expression, whether written explicitly or understood. Like: A quantity saying how many copies to use.
Reciprocal: Pronounced ree-SIP-ruh-kul. One divided by a nonzero number; for a nonzero fraction, exchange top and bottom. Like: Turning a fraction upside down.
Ratio: Pronounced RAY-shee-oh. One quantity divided by another, comparing their sizes. Like: Miles traveled for each hour spent traveling.
Interval: Pronounced IN-ter-vul. A connected stretch of a number line, with notation indicating whether its endpoints are included. Like: A portion of a road between two mile markers.
Square root: Pronounced skwair root. The nonnegative number which, multiplied by itself, gives the number inside the root sign. Like: Finding a square tile's side from its area.
Substitution: Pronounced sub-stih-TOO-shun. Replacing a letter by a specific value in an equation or expression. Like: Putting your proposed key into the lock to check it.
Fundamental period: Pronounced fun-duh-MEN-tul PEER-ee-ud. The smallest positive repeat distance; this is what period means for the nonconstant waves here. Like: One wallpaper tile's width rather than several tiles together.
Quarter turn: Pronounced KWOR-ter turn. One fourth of a full circle turn, equal to 90° or $π over 2$ radians. Like: Moving a clock hand through one fourth of its face.
Quadrant: Pronounced KWOD-runt. One of the four regions made by the coordinate axes, numbered counterclockwise from the upper right. Like: One room in a four-room floor plan.
Right triangle: Pronounced rite TRY-ang-gul. A triangle with one 90° angle. Its other two angles together use the remaining 90°. Like: A triangular tile with one square corner.
Acute angle: Pronounced uh-KYOOT ANG-gul. An angle greater than 0° and less than 90°. Like: A partly opened door before it reaches a square corner.
Hypotenuse: Pronounced hy-POT-uh-noos. The longest side of a right triangle, across from its 90° angle. Like: The sloping edge across from a square corner.
Opposite side: Pronounced OP-uh-zit side. The right triangle leg across from the acute angle you chose; it does not touch that angle. Like: The wall across the room from your chosen corner.
Adjacent side: Pronounced uh-JAY-sunt side. The right triangle leg touching the acute angle you chose. The hypotenuse also touches it but is not the adjacent leg. Like: The wall beside your chosen corner.
Reference angle: Pronounced REF-er-uns ANG-gul. The acute angle between an angle's terminal ray and the nearest horizontal axis; it helps find coordinate magnitudes. Like: Measuring a ray's small tilt from the horizontal.
Clockwise: Pronounced KLOK-wise. Turning in the direction a clock's hands move; from the rightmost circle point, the first motion is downward. Like: Following the hands of an ordinary clock.
Counterclockwise: Pronounced kown-ter-KLOK-wise. Turning opposite a clock's hands; from the rightmost circle point, the first motion is upward. Like: Following a clock's hands backward.
Increasing: Pronounced in-KREE-sing. A graph rises as its input moves right through the interval being discussed. Like: Walking uphill while moving forward.
Decreasing: Pronounced dee-KREE-sing. A graph falls as its input moves right through the interval being discussed. Like: Walking downhill while moving forward.
Pythagorean theorem: Pronounced py-thag-uh-REE-un THEER-um. In a right triangle, the squares of the two leg lengths add to the square of the hypotenuse length. Like: Combining two smaller square areas into one larger square area.
SOH CAH TOA: Pronounced soh kah toh-uh. A memory phrase for sine as opposite over hypotenuse, cosine as adjacent over hypotenuse, and tangent as opposite over adjacent. Like: Three short labels that recall three side comparisons.
Degree: Pronounced duh-GREE. An angle unit defined by convention as one of 360 equal parts of a full turn. The ° sign distinguishes degrees from radians. Like: One of 360 equal marks around a circular dial.
Radius: Pronounced RAY-dee-us. The distance from a circle's center to its rim. Like: A spoke running from a wheel's center to its edge.
Ray: Pronounced ray. A line starting at one endpoint and extending forever in one direction. Like: A beam sent outward from a lamp.
Initial side: Pronounced ih-NISH-ul side. The ray where an angle's turn begins. Like: The direction you face before turning.
Terminal side: Pronounced TER-mih-nul side. The ray where an angle's turn finishes. Like: The direction you face after turning.
Vertex: Pronounced VER-teks. The shared endpoint of an angle's two rays. Like: The hinge where a door turns.
Numerator: Pronounced NOO-muh-ray-ter. The top of a fraction. Like: The number of equal slices you are counting.
Denominator: Pronounced dih-NOM-uh-nay-ter. The bottom of a fraction; it must be nonzero. Like: The number of equal pieces making a whole.
Rationalizing: Pronounced RASH-uh-nuh-ly-zing. Rewriting an equivalent fraction to remove a square root from its denominator. Like: Repackaging the same amount with a different label.
Set-builder notation: Pronounced set BIL-der noh-TAY-shun. Braces collect values meeting the condition after the bar, which means such that. Like: A guest list defined by a requirement.
Phase shift: Pronounced fayz shift. The horizontal displacement of a wave from its chosen parent graph. Like: Starting the same recording earlier or later.

Quick checks

Find the amplitude and period of y = −5 cos(4x).

Amplitude = 5
Period = $π over 2$ , because |−5| = 5 and $2 π over 4$ = $π over 2$ .

Find the period of y = 3 sin(x).

4, because 2π ÷ = 2π × = 4.

A zero-midline wave has maximum 7, minimum −7, period π, and passes through the middle going up at x = 0. Write an unshifted equation using positive ω.

y = 7 sin(2x), because the half-height is 7, = 2, and rising through the middle chooses positive sine.

Fill the blank: cos 20° = sin ___.

70°, because 90° − 20° = 70° and sine is cosine's cofunction.

Name the five cosine key points on [0, 2π].

(0, 1)
( $π over 2$ , 0)
(π, −1)
( $3 π over 2$ , 0)
(2π, 1), because cosine follows top, middle, bottom, middle, top.

Is sine odd or even, and what symmetry does its graph have?

Odd
Symmetry about the origin, because sin(−x) = −sin x changes both coordinates of a graph point.

Why is the full domain of y = sin x different from its one-cycle drawing interval [0, 2π]?

The domain is every real number, because any signed turn is allowed. [0, 2π] is only one complete piece of the repeating picture.

Does the constant graph y = 0 have smallest positive period 2π?

No. Every positive shift repeats a constant, so there is no smallest positive period. The sine/cosine period formula requires A ≠ 0 and ω ≠ 0.

Before you start

Explain it like I am five: record a turning wheel
Picture a wheel with a dot on its rim. Start with the dot at the right side. As the wheel turns, the dot rises, reaches the top, falls past the middle, reaches the bottom, and returns to the right side.
Now let a strip of paper move sideways while you record how high the dot is. The wheel goes around in a circle, but the height record makes a wave. The horizontal position on the paper tells you how far the wheel has turned.
A taller wheel makes a taller height record. A wheel that turns faster fits more repeats into the same length of paper. Sine records the height, and cosine records the sideways position. This section teaches how to draw and measure those records. Here radius means distance from the center to the rim. A unit circle has radius 1. The letter x names the turn and y names the recorded height. sin means sine and cos means cosine.
Read the symbols, fractions and squares before using them
Think of math notation as labels on measuring tools. A letter names the quantity you are measuring or the rule you are using. A fraction counts equal pieces of a whole. A square repeats a multiplication. None of these marks is a new instruction to guess. Here you learn how to say them and what calculation each asks for, before the circle and triangle examples use them. π, pronounced pie, is a particular positive number, approximately 3.14; it is not an angle unit or an instruction. Radians and degrees are angle units. A degree, written °, is one of 360 equal pieces of a full turn. This convention gives everyone the same unit for measuring turns. The letter θ, pronounced thay-tuh, names an angle.
Input, output, coordinates and graph paper
An Input is the number you give a rule. An Output is the number the rule returns. A vending machine accepts your choice and returns an item. A Trigonometric function accepts an angle and returns a number. On graph paper, a Coordinate tells how far to move along one axis. An Ordered pair (x, y) gives two coordinates in order. The Graph of a function shows its input and output together in the xy-plane, the flat plane containing the x-axis and y-axis. The x-axis runs horizontally and the y-axis vertically; an axis is a reference line for locating positions. The Origin is (0, 0), their meeting point. A variable is a letter whose value can change. f is a name for a rule, and f(x) means its output at input x, pronounced f of x. The parentheses here hold the input; f(x) does not mean f times x. In arithmetic, however, 3x means 3 × x. An equation uses = to say that its two sides have equal values.
Radians, real numbers and the unit circle
A Radian measures a turn using distance along a circle. On the Unit circle, whose radius is 1, an Arc length of 1 produces a turn of 1 radian. A Central angle has its corner at the circle's center. A full lap has length 2π, so a full turn is 2π radians. A Real number can be a whole number, a fraction, a negative number, or an endless decimal such as π. Every real number can describe a forward or backward walk around this circle. An angle measures the turn from a starting direction to a finishing direction. Each direction can be drawn as a Ray, a line starting at the center and extending outward without an end. The starting ray is the Initial side; the finishing ray is the Terminal side. Their shared starting point is the Vertex. Unless another starting direction is named, you start along the horizontal axis pointing right. In θ = $\frac{s}{r}$ , θ names the angle in radians, s names the signed curved rim distance, and r names the positive radius. Ordinary arc length is nonnegative; signed rim distance also records the direction of travel.
Absolute value and signed arithmetic
Absolute value is a distance from 0, so it has no negative sign. |−6| = 6 and |6| = 6. Multiplying by a negative number reverses the direction of a signed quantity. Two reversals undo each other, so a negative times a negative is positive. Subtracting a negative means undoing a backward move, which is a forward move. These ideas explain why a reflected wave can have a negative coefficient and a positive amplitude.
Multiply and divide fractions, including π
Multiplying fractions means taking a fraction of a fraction. Multiply top by top and bottom by bottom. Dividing asks how many copies of one amount fit into another. Dividing by a fraction means multiplying by its reciprocal, the fraction turned upside down. A nonzero common factor can be canceled from top and bottom because dividing both by the same number keeps the quotient unchanged. π behaves like a nonzero number in this arithmetic. The numerator is the top of a fraction, and the denominator is its bottom. A denominator counts equal pieces and must be nonzero. A factor is a whole quantity multiplied by another quantity. For example, 5π means 5 × π, so 5 and π are factors. To reduce $10 over 12$ , divide top and bottom by 2 to get $5 over 6$ ; both fractions name the same amount.
Subtract angle fractions using a common denominator
You can subtract slices only when they have the same size. A half and a third use different slice sizes, so rewrite them as sixths before subtracting. Fractions of π work the same way. In a cofunction calculation, you subtract an angle from a quarter turn. Degrees must be subtracted from degrees and radians from radians, the way inches must be compared with inches. A denominator is the number on the bottom. A common denominator gives the two fractions equal slice sizes without changing either amount. In this refresher, quarter turn means one fourth of a full 360° turn, so it is 90°, or $π over 2$ radians. Cofunction is the name for a partner trig function; lesson 5 will explain why partner arguments add to a quarter turn. Here you are learning only the subtraction needed for that later method.
Solve a linear equation and plug the answer back in
An equation says that two quantities are equal, like two balanced pans. To find a missing number, undo the operation hiding it. If a number was multiplied by 5, divide by 5. Do the same operation to both sides so the balance stays level. A letter such as T or ω is a place for the missing number. Substitution means putting your proposed value back in the original equation to see whether it really works. Linear equation here means an equation where the missing letter is multiplied by a fixed number or has a fixed number added, without powers of that letter. A positive Coefficient is a positive multiplier. The letter ω, called omega and pronounced oh-MAY-guh, is a multiplier; T will later name a cycle's input length. A cycle is one full repeat of a pattern. You can solve their equations without knowing the later graph method yet.
Sine, cosine and exact special-angle values
Picture a wheel dot after a turn. Its sideways position and height give two measurements, like an address with a street position and a floor. The letter θ, pronounced thay-tuh, names the turn. P names the circle point; P(θ) means that point after turn θ. The point on the unit circle after a turn θ is (cos θ, sin θ). Cosine gives its sideways position and Sine gives its height. Do not confuse these circle coordinates with a wave graph's input and output. A wave graph uses the angle as its horizontal input. Exact value means a value without rounding. The two special right triangles let you rebuild sine and cosine values as fractions and square roots instead of memorizing an unexplained decimal table. A Square root $square root of 9$ is the nonnegative number whose square is 9: 3, because 3 × 3 = 9. Thus $square root of 3$ × $square root of 3$ = 3 even though $square root of 3$ has no terminating decimal. A Right triangle has one 90° corner. An Acute angle is between 0° and 90°. The Hypotenuse is across from the right angle and is the longest side; the other two sides are legs. The Opposite side is the leg across from the acute angle you chose. The Adjacent side is the leg touching that angle. Changing your chosen angle exchanges those two jobs.
Domain, range, interval notation and rounded decimals
Domain means allowed inputs. Range means outputs that actually occur. An interval describes a stretch of a number line. Square brackets include an endpoint, while parentheses exclude it. The symbols −∞ and ∞ describe continuation without an end, rather than numbers you can include. An Exact value keeps the full number, such as π. A Rounded decimal keeps a chosen number of places, such as π ≈ 3.14. The sign ≈ means approximately equal. The sign < means less than, > means greater than, ≤ means less than or equal to, and ≥ means greater than or equal to. For example, −1 ≤ y ≤ 1 means y is at least −1 and at most 1. In {x | x is a real number}, the braces collect the allowed values and the bar means such that. This is called Set-builder notation. In {1}, the braces collect only the value 1.
The four other trig names and their fractions
Tangent, Cotangent, Secant and Cosecant are built from sine and cosine. A reciprocal is a number turned upside down, or 1 divided by that number. A ratio means one amount divided by another. Tangent compares height with sideways position, so it is sine divided by cosine. Cotangent reverses that comparison. Secant is the reciprocal of cosine, and cosecant is the reciprocal of sine. A denominator of zero makes a value Undefined: there is no number which multiplied by zero gives a nonzero amount.