Graphs of the other four functions

You will first find which angle inputs each of the four functions accepts and which output heights it can produce. Then you will use symmetry and period to recognize repeated pieces of the graphs. Tangent will show you why division by a number near zero creates a vertical asymptote, and cotangent will use the same reasoning with a different denominator. Finally, you will build cosecant from sine and secant from cosine, then practice sketching and checking all four without a calculator.

Lessons

Vocabulary

Trigonometric function trig-uh-nuh-MET-rik FUNK-shun: A function whose input is an angle and whose output comes from circle coordinates or their ratios. Like: A number machine with an angle dial.
Tangent TAN-jent: Sine divided by cosine, defined when cosine is nonzero. Its graph has rising branches. Like: A ramp's rise divided by its run.
Cotangent koh-TAN-jent: Cosine divided by sine, defined when sine is nonzero. Its graph has falling branches. Like: A ramp's run divided by its rise.
Secant SEE-kant: One divided by cosine, defined when cosine is nonzero. Its graph has y-axis symmetry. Like: Count how many cosine-sized pieces fit in one whole.
Cosecant koh-SEE-kant: One divided by sine, defined when sine is nonzero. Its graph has origin symmetry. Like: Count how many sine-sized pieces fit in one whole.
Sine sine: The vertical coordinate of the unit-circle point for an angle. Like: The height of a seat on a wheel, relative to its center.
Cosine KOH-sine: The horizontal coordinate of the unit-circle point for an angle. Like: How far a wheel's seat is to the right or left of its center.
Unit circle YOO-nit SUR-kul: A circle of radius 1 centered at the origin. Its points have coordinates (cos x, sin x). Like: A round track one step from its center pole.
Quadrantal angle kwah-DRAN-tul ANG-gul: An angle whose terminal side lies on an axis. It is an integer multiple of $π over 2$ radians, equivalently an integer multiple of 90°. Like: A clock hand pointing exactly right, up, left or down.
Radian RAY-dee-un: An angle unit based on arc length divided by radius. A half-turn is π radians, equal to 180°. Like: Measuring a wheel's turn by distance walked around its rim.
Integer IN-tuh-jer: A whole count, including positive counts, negative counts and zero. Like: Numbered stops extending in both directions from home.
Multiple MUL-tuh-pul: A number obtained by multiplying a given number by an integer. Like: Equally spaced stops reached by a repeated step.
Domain doh-MAYN: The set of inputs for which a function has an output. Like: The choices a machine accepts.
Range raynj: The set of output values a function actually reaches. Like: The heights marked by a graph's shadow on a wall.
Real number REE-ul NUM-ber: A number on the ordinary number line, including integers, fractions and square roots of nonnegative numbers. Like: Any location along a straight measured road.
Undefined un-duh-FINED: No output exists at that input. Here the formula would divide by zero. Like: An input button that the machine cannot accept.
Numerator NOO-muh-ray-ter: The top of a fraction, the amount being divided. Like: The amount you want to share.
Denominator duh-NOM-uh-nay-ter: The bottom of a fraction, the number you divide by. It must be nonzero. Like: The size of each piece when counting pieces in a whole.
Zero ZEER-oh: For a function, an allowed input where the output is 0. A zero of the denominator instead creates a forbidden input. Like: A place where the graph meets ground level.
x-intercept EKS IN-ter-sept: A graph point on the horizontal axis, where the output is zero. Like: A road crossing ground level.
y-intercept WHY IN-ter-sept: A graph point on the vertical axis, where the input is zero. Like: The height recorded at the starting input.
Reciprocal rih-SIP-ruh-kul: One divided by a nonzero number. A number times its reciprocal equals 1. Like: Turn a fraction upside down.
Reciprocal identities rih-SIP-ruh-kul eye-DEN-tuh-teez: Equalities linking sine with cosecant and cosine with secant. Cotangent and tangent are reciprocal partners wherever both sides are defined. Like: Pairs of fractions turned upside down.
Quotient identities KWOH-shunt eye-DEN-tuh-teez: The formulas tan x = sine divided by cosine and cot x = cosine divided by sine, with nonzero denominators. Like: Two coordinates compared as rise over run or run over rise.
Reciprocal graph rih-SIP-ruh-kul graf: A graph made by keeping each input and replacing every nonzero output y with 1 divided by y. Like: A machine changes a sign's height but leaves its address.
Vertical asymptote VER-tih-kul AS-im-toht: A vertical line approached by inputs whose output magnitudes become unbounded from at least one side. Like: A fence beside a ramp that climbs beyond any chosen height.
Not continuous (discontinuous) not kun-TIN-yoo-us; dis-kun-TIN-yoo-us: A graph with a break. These four full graphs have breaks at excluded angles, while each branch is continuous. Like: Separate road stretches divided by closed gaps.
Continuous kun-TIN-yoo-us: Having no break within an allowed interval. Each branch here is continuous even though excluded inputs separate branches. Like: An unbroken stretch of road.
Branch branch: One connected piece of a graph. Here consecutive asymptotes bound each branch. Like: One stretch of road between two closed gates.
Unbounded un-BOWN-did: Having no finite ceiling or floor in the specified direction. Outputs eventually pass any proposed bound. Like: Climbing beyond every height someone names.
Infinity in-FIN-ih-tee: A symbol describing unlimited continuation, not a real output or an included endpoint. Like: A road with no final mile marker.
Period PEER-ee-ud: A positive horizontal input distance after which the whole pattern repeats. Like: The width of a repeating wallpaper design.
Fundamental period fun-duh-MEN-tul PEER-ee-ud: The smallest positive period of a function. Like: The shortest whole wallpaper design that repeats.
Periodic function peer-ee-OD-ik FUNK-shun: A function whose outputs and allowed-input pattern repeat after a fixed positive input distance. Like: A repeating wallpaper strip.
Even function EE-vun FUNK-shun: A function whose domain includes opposite inputs together and has f(−x) = f(x). Its graph has y-axis symmetry. Like: A picture matching its vertical mirror image.
Odd function od FUNK-shun: A function whose domain includes opposite inputs together and has f(−x) = −f(x). Its graph has origin symmetry. Like: A picture unchanged after a half-turn about its center.
Symmetry SIM-uh-tree: A picture matches itself after a specified reflection or rotation. Like: Two matching sides of a folded design.
With respect to (w.r.t.) with rih-SPEKT too: In relation to a specified reference. For symmetry, it names the axis or point used. Like: A mirror location determines which sides are compared.
x-axis EKS AK-sis: The horizontal coordinate line, whose points have output coordinate y = 0. Like: The east-west road on a map.
y-axis WHY AK-sis: The vertical coordinate line, whose points have input coordinate x = 0. Like: The north-south road through the map's starting point.
Origin OR-ih-jin: The crossing of the axes, with coordinates (0, 0). Like: The start of two measuring roads.
Interval notation IN-ter-vul noh-TAY-shun: A way to describe a continuous span using endpoints. Brackets include finite endpoints; parentheses exclude them. Like: Write the accepted stretch between two road signs.
Union YOON-yun: Joining two sets: a value in either set is allowed. The symbol is ∪. Like: Two permitted road stretches on one map.
Amplitude AM-plih-tood: A finite half-height between a bounded wave's highest and lowest outputs. None of these four whole graphs has a finite amplitude. Like: Half the total up-and-down travel of a bounded wave.
Increasing in-KREE-sing: Within an interval, a larger input gives a larger output. Like: Walking right along a rising road.
Decreasing dih-KREE-sing: Within an interval, a larger input gives a smaller output. Like: Walking right along a descending road.
Vertex VER-teks: An angle's corner, where its rays start, or a reciprocal branch's turning point closest to the x-axis. The surrounding problem tells you which meaning applies. Like: A clock hand's pivot or the turning point of a curved bowl.
One-sided approach wun SY-did uh-PROHCH: Moving toward an input while keeping every nearby input entirely on its left or entirely on its right. Like: Approach a doorway from one direction along a hallway.
Coordinate koh-OR-duh-nit: One number in a point's address, recording a horizontal or vertical position. Like: One part of a two-step walking instruction.
x-coordinate EKS koh-OR-duh-nit: The first coordinate of a point. On a function graph it gives the input's horizontal position. Like: How far east or west to walk first.
y-coordinate WHY koh-OR-duh-nit: The second coordinate of a point. On a function graph it gives the output height. Like: How far up or down to walk second.
Magnitude MAG-nih-tood: A number's distance from zero, ignoring its sign. Like: How far a location is from home, either left or right.
Absolute value AB-suh-loot VAL-yoo: A number's distance from zero, also called its magnitude. Vertical bars indicate it. Like: Distance from home regardless of the direction traveled.
Variable VAIR-ee-uh-bul: A letter standing for a number or an input that can vary. Like: A blank address slot filled by the chosen number.
Function notation FUNK-shun noh-TAY-shun: A name followed by an input in parentheses, recording the output of that rule at that input. Like: Label the machine, then name the button pressed.
Ratio RAY-shee-oh: One number compared with another by division, with a nonzero divisor. Like: A ramp's rise per unit of run.
Ray ray: A line that starts at one point and extends without end in one direction. Like: A beam leaving a flashlight.
Initial side ih-NISH-ul side: The starting ray of an angle's turn. Like: A clock hand's direction before it turns.
Terminal side TUR-mih-nul side: The ending ray of an angle's turn. Like: A clock hand's direction after the turn.
Standard position STAN-derd puh-ZISH-un: An angle placed with its vertex at the origin and its initial side pointing along the positive x-axis. Like: Always begin a route from the same place and direction.
Quadrant KWAH-drunt: One of the four regions between the coordinate axes, numbered counterclockwise from upper right. Like: One region around two crossing roads.
Reflection rih-FLEK-shun: A mirror move across a line that reverses the perpendicular coordinate and keeps the coordinate along the line. Like: A location and its mirror image across a road.
Radius RAY-dee-us: The distance from a circle's center to any point on its rim. Like: The length of a wheel's spoke.
Arc ark: A curved piece of a circle's rim. Like: A stretch of the track around a wheel.
Right angle rite ANG-gul: A quarter-turn angle, 90° or $π over 2$ radians. Like: The square corner of a tile.
Right triangle rite TRY-ang-gul: A triangle with one right angle. Like: Two perpendicular walks and the direct return path.
Leg leg: Either side of a right triangle that meets at its right angle. Like: One of two perpendicular legs of a walking route.
Hypotenuse hy-POT-uh-noos: The side across from the right angle in a right triangle. Like: The direct diagonal across two perpendicular walks.
Equilateral triangle ee-kwuh-LAT-er-ul TRY-ang-gul: A triangle with three equal sides and three 60° angles. Like: A triangle with three matching edges.
Pythagorean theorem py-thag-uh-REE-un THEE-uh-rum: In a right triangle, the squares of the leg lengths add to the square of the hypotenuse. Like: Two smaller square areas equal the area on the diagonal.
Set-builder notation set BIL-der noh-TAY-shun: Braces surrounding a variable and a condition, read as all values of the variable such that the condition holds. Like: An admission rule specifying who belongs to a collection.

Quick checks

Where are the vertical asymptotes of y = cot x?

x = nπ for any integer n: …, −π, 0, π, 2π, …. Sine is zero there, so cotangent's denominator vanishes.

Which of the four functions are odd?

Tangent, cotangent and cosecant. Each reverses its output when the input changes sign
secant keeps the output and is even.

What is the range of y = sec x?

(−∞, −1] ∪ [1, ∞). The reciprocal of a nonzero cosine between −1 and 1 has size at least 1.

Why does y = tan x have no amplitude?

Amplitude is a finite half-height between a highest and lowest output. Tangent has neither extreme because its range is all real numbers.

Is cot() zero or undefined?

Zero: cos() = 0 and sin() = 1, so cot() = = 0. The zero is only on the top.

What does a vertical asymptote tell you?

At least one side has output magnitudes that grow without bound as the input approaches that line. For these four parents the line's input itself is undefined.

What are the fundamental periods of tangent and cosecant?

Tangent: π, because two coordinate sign changes cancel.
Cosecant: 2π, because one denominator sign change after π reverses its output.

Does secant's range include −1? Does it include 0?

−1 is included because sec π = −1.
0 is excluded because a nonzero cosine reciprocal cannot equal zero.

Before you start

Explain it like I am five
Picture a number machine with a button marked divide 1 by the input. If you enter a tenth, ten tenths fit into one whole, so the machine returns 10. Enter a hundredth and it returns 100. Smaller pieces mean more pieces fit.
At zero the machine has no answer. No number of zero-sized pieces can add up to one. The machine cannot return zero or infinity; that input has no output at all.
Now draw a dot for each accepted input and its output. As a positive input gets close to zero, the dots climb beyond any height you choose. A vertical guide line marks the input they approach. The four graphs in this section use the same division idea, with sine or cosine supplying the number on the bottom.
Coordinates: turn an input and output into a point
A graph is a map of a function's outputs. A function is a rule that gives one output for each allowed input. The point (x, y) gives you directions: move x along the horizontal line, then move y vertically. The x-axis is the horizontal line where y = 0. The y-axis is the vertical line where x = 0. Their crossing, (0, 0), is the origin. On a trig graph x is an angle input, usually in radians, and y is the value of the function. A real number is a location on the ordinary number line, including a whole number, fraction or exact square root. A variable is a letter standing for a number. The letter f names a function, and f(a) means the output when its input is a; parentheses here mean enter the input. Radians measure angles: one full turn is 2π radians, so π is a half-turn and $π over 4$ is one eighth of a full turn. You will practice those angle units below.
Fractions, signs and reciprocals
A fraction is a division: the numerator is the top and the denominator is the bottom. $3 over 4$ means 3 divided by 4. A reciprocal is 1 divided by a nonzero number. For a fraction you get its reciprocal by turning the top and bottom around. The reciprocal of $3 over 4$ is $4 over 3$ . This is a change in size, not a change of sign. A negative number has a negative reciprocal. A zero numerator is allowed when the denominator is nonzero; a zero denominator is never allowed. Think of a whole cut into equal slices. To multiply fractions, multiply their numerators and multiply their denominators. To reduce a fraction, divide its top and bottom by the same nonzero factor. A factor is a number multiplied as part of a product.
Square roots and exact reciprocal values
Think of square floor tiles. A square with side length 2 has area 2 × 2 = 4. The notation $a^{2}$ means a × a. A square root reverses that question: what nonnegative side length, meaning a length zero or greater, has the stated square? Thus $square root of 2$ is the positive number whose square is 2, and $square root of 0$ = 0. The number $square root of 2$ is not a whole number, so you keep $square root of 2$ in an exact answer. To rationalize a denominator means to rewrite a fraction with a root on the bottom so the bottom has no root. This changes the form, not the value. Multiply top and bottom by the same nonzero number, which is multiplying by 1. This turns the reciprocal of $square root of 2 over 2$ into $square root of 2$ .
Radians, integer multiples and fractions of π
A radian is a unit for an angle. One full turn is 2π radians and a half-turn is π radians, the same angle as 180°. You can therefore replace π by 180° to convert a fraction of π to degrees. An integer is a whole count that may be positive, negative or zero. A multiple of π is an integer times π. In x = nπ, n names every integer, so the formula includes angles to the left of zero as well as to the right. In x = $π over 2$ + nπ, the same steps start from $π over 2$ . Picture walking around the rim of a wheel. One radian turns through an arc, a piece of the rim, whose length equals the radius, the distance from the center to the rim. Angle in radians is arc length divided by radius. A full rim has length 2π times its radius, so a full turn is 2π radians. The number π is the circle's circumference, the distance all the way around its rim, divided by its diameter, approximately 3.141593. A diameter goes across the center and is twice the radius.
Angle regions and circle directions
Picture a clock hand turning around the crossing of two roads. Its starting direction is its initial side, and its ending direction is its terminal side. Those directions are rays, lines that start at the crossing and extend in one direction. For standard position, start at the origin and point right along the positive x-axis. Counterclockwise turns are positive by convention; clockwise turns are negative. The axes divide the plane into four quadrants, regions numbered I, II, III and IV. This lets you predict coordinate signs before dividing anything. The two angle rays share a starting point called the angle's vertex, or corner. This is a different use of vertex from a graph branch's turning point, which you will meet later.
Inequalities, intervals and joining two allowed pieces
An inequality compares sizes. y ≥ 1 says y is 1 or more, and y ≤ −1 says y is −1 or less. Interval notation gives the same information with endpoints. A square bracket includes the endpoint; a parenthesis excludes it. Infinity means the line continues without an end, so it is always next to a parenthesis. Union, written ∪, joins two allowed pieces. It means you may be in the first piece or the second. This will describe the missing middle of the secant and cosecant output heights. The symbols < and > mean less than and greater than. Adding an equals sign allows equality: ≤ means less than or equal to, and ≥ means greater than or equal to. A set is a collection of values. The symbol ∈ means belongs to. Set-builder notation states the condition for belonging: {y | y ≥ 1} means all real y such that y is at least 1; the bar means such that.
Solving an equation and checking the result
To solve an equation is to find every input that makes its two sides equal. Think of a balance scale: you can make the same change on both sides while keeping the balance. For 2x − π = 0, solving locates the input that makes the expression 2x − π equal zero. A substitution check puts the answer back into the original expression. This check tells you whether you found the right input. When we solve a denominator equation later, the purpose is to find an input to exclude, not an output to plot. Isolate the variable means leave its letter by itself on one side.
Circle distances and the two special triangles
Picture a wheel of radius 1, with a spoke reaching a point above and right of the center. Walk horizontally and then vertically to that point. Those two walks and the spoke form a right triangle. A right angle is a quarter-turn, 90°. The two sides meeting at that angle are its legs. The side across from it is the hypotenuse. The Pythagorean theorem says the squares of the leg lengths add to the square of the hypotenuse. This connects a circle's coordinates with exact triangle lengths. You can rebuild the 45° and 30° points instead of memorizing unexplained roots. Perpendicular means meeting at a right angle, as the horizontal and vertical walks do. The Greek letter θ, pronounced theta, is an angle label in a triangle picture. It plays the same role as x when x names an angle.
Sine and cosine: read the unit-circle point
The Unit circle is a circle of radius 1 centered at the origin. Start at (1, 0), then turn counterclockwise for a positive angle and clockwise for a negative angle. Cosine is the first coordinate of the stopping point, the horizontal one. Sine is the second coordinate, the vertical one. At axis angles you can read both coordinates without any triangle. At a 45° angle the horizontal and vertical lengths match; at 30° and 60° they come from half an equilateral triangle. These are the few values used to anchor the new graphs.