Properties of the trigonometric functions

Lessons

1. Sine and cosine: every input works, and the output stays between −1 and 1I can explain this
2. Quadrantal angles: where tan, cot, sec and csc breakI can explain this
3. The range of tan, cot, sec and cscI can explain this
4. Periodic functions: the values repeat every lapI can explain this
5. Signs of the six functions in the four quadrantsI can explain this
6. Which quadrant is the angle in?I can explain this
7. Increasing and decreasing: where sine and cosine rise and fallI can explain this
8. Even and odd: what a minus sign inside doesI can explain this
9. Combining properties to find signs and known exact valuesI can explain this
10. All six functions from one value and a quadrantI can explain this

Vocabulary

Domain DOH-mayn

All the inputs a function accepts. For the trig functions, the inputs are angles, or real numbers t read as radians. Like: The coins a vending machine will accept.

Range raynj

All the outputs a function can actually produce. Like: The snacks the vending machine can actually drop.

Interval notation IN-ter-vul noh-TAY-shun

A short way to write a stretch of numbers: [ ] includes an endpoint, ( ) leaves it out, and ∞ always takes ( ). Like: A guest list saying 'from Ann through Zoe' versus 'after Ann, before Zoe'.

Union (∪) YOO-nyun

The symbol ∪ joins two sets of numbers; a number belongs to the union if it is in either piece. Read it as 'or'. Like: Two parking lots under one sign: you may park in either.

Integer n IN-tuh-jer en

An integer is a whole number, including negatives and zero. In t ≠ π2$\frac{\pi }{2}$ + nπ, n ranges over every integer, so the rule excludes infinitely many inputs. Like: Floor numbers in a building with basements: 2, 1, 0, −1, −2.

Unit circle (U) YOON-it SER-kul

The circle of radius 1 centered at the origin. Every point (x, y) on it satisfies x²$x^2$ + y²$y^2$ = 1. The notes call it U. Like: A round running track exactly 1 unit from the center everywhere.

The input t in-put tee

On U, t is an angle in radians and the directed distance traveled from (1, 0). The notes reserve x for the point's first coordinate. Unsigned traveled length is |t|. Like: Meters run on a track of radius 1: the distance and the angle match.

Terminal point P(cos t, sin t) TER-muh-nul point

The point where you stop after walking a distance t along the unit circle from (1, 0). Its x-coordinate is cos t and its y-coordinate is sin t. Like: The spot where a runner stops after t meters on the track.

Quadrant KWAH-drunt

One of the four regions the two axes cut the plane into, numbered I, II, III, IV counterclockwise from the upper right. An angle is in the quadrant that holds its terminal side. Like: The four slices of a pizza cut once across and once down.

Quadrantal angle kwah-DRAN-tul ANG-gul

An angle whose terminal side lies on an axis: a multiple of π2$\frac{\pi }{2}$ (90°), such as 0, ±π2$\frac{\pi }{2}$, ±π, ±3π2$\frac{3\pi }{2}$, 2π. It belongs to no quadrant. Like: A clock hand pointing exactly at 12, 3, 6 or 9.

Undefined un-duh-FYND

A function is undefined at an input when its formula would divide by zero there. No output exists, so that input is left out of the domain. Like: Sharing 6 cookies among 0 friends: the question has no answer.

Periodic function peer-ee-AH-dik FUNK-shun

A function that repeats after a positive shift k: shifting preserves its domain and f(t + k) = f(t) for every allowed t. Like: The days of the week: every 7 days it is Monday again.

Period PEER-ee-ud

The least positive repeat length, when one exists. The six trig functions have least periods; a constant function repeats after any positive shift and has no least positive period. Like: The 7 in 'every 7 days'.

Coterminal angles koh-TER-muh-nul ANG-gulz

Angles that end on the same terminal side. They differ by whole turns: a multiple of 360°, or of 2π. Like: Two runners who stop at the same spot, one after an extra lap.

Coterminal reduction koh-TER-muh-nul ree-DUK-shun

Replacing an angle by its coterminal angle in [0°, 360°) or [0, 2π) before finding its quadrant or its value. The notes write 'co-terminal'. Like: Reading '30 hours after midnight' as 6 AM the next day.

Increasing function in-KREE-sing FUNK-shun

A function is increasing on an interval if a larger input always gives a larger output there: x₂$x_2$ > x₁$x_1$ means f(x₂$x_2$) > f(x₁$x_1$). Like: Walking uphill: every step forward takes you higher.

Decreasing function dee-KREE-sing FUNK-shun

A function is decreasing on an interval if a larger input always gives a smaller output there: x₂$x_2$ > x₁$x_1$ means f(x₂$x_2$) < f(x₁$x_1$). Like: Walking downhill: every step forward takes you lower.

Even function EE-vun FUNK-shun

A function with a domain symmetric about 0 and f(−x) = f(x) for every allowed x. Opposite inputs give equal outputs; the graph is symmetric about the y-axis. Like: A face that looks the same in a mirror.

Odd function ahd FUNK-shun

A function with a domain symmetric about 0 and f(−x) = −f(x) for every allowed x. Opposite inputs give opposite outputs; the graph is symmetric about the origin. Like: A pinwheel that looks the same after half a turn.

Symmetric about the y-axis SIM-uh-trik uh-BOUT thuh why AK-sis

The left half of the graph is the mirror image of the right half, folded along the y-axis. The notes write 'symmetric w.r.t. the y-axis', w.r.t. meaning 'with respect to'. Like: The two wings of a butterfly.

Symmetric about the origin SIM-uh-trik uh-BOUT thee OR-uh-jin

Turning the graph half a turn (180°) about the origin leaves it unchanged: every point (a, b) on it has a partner (−a, −b). Like: A playing card that reads the same upside down.

Reciprocal identities rih-SIP-ruh-kul eye-DEN-tuh-teez

csc θ = 1sin θ$\frac{1}{\sin \theta }$, sec θ = 1cos θ$\frac{1}{\cos \theta }$, cot θ = 1tan θ$\frac{1}{\tan \theta }$. Denominators must be nonzero and both sides defined. Partners have the same sign on their shared domain. Like: Miles per hour and hours per mile: each is the other flipped.

Quotient identities KWOH-shunt eye-DEN-tuh-teez

tan θ = sin θcos θ$\frac{\sin \theta }{\cos \theta }$ and cot θ = cos θsin θ$\frac{\cos \theta }{\sin \theta }$: tangent and cotangent are ratios of sine and cosine. Like: The slope of a ramp: rise divided by run.

Pythagorean identities pih-THAG-uh-REE-un eye-DEN-tuh-teez

sin²${\sin }^2$θ + cos²${\cos }^2$θ = 1 always. tan²${\tan }^2$θ + 1 = sec²${\sec }^2$θ requires cos θ ≠ 0; cot²${\cot }^2$θ + 1 = csc²${\csc }^2$θ requires sin θ ≠ 0. sin²${\sin }^2$θ means (sin θ)²${)}^2$. Like: The ladder rule: height²$t^2$ + reach²$h^2$ = ladder²$r^2$, with a ladder of length 1.

LHS and RHS el-aych-es and ar-aych-es

LHS means left side and RHS means right side of an equation. Verifying an identity means transforming one side into the other for every input where the original expression exists. Like: The two pans of a balance scale you are checking.

Exact value ig-ZAKT VAL-yoo

A value stated without rounding, often written using whole numbers, fractions or square roots. An exact expression keeps the complete value instead of stopping at a rounded decimal. Like: Saying 'one third' instead of '0.33'.

Rationalize the denominator RASH-uh-nuh-lyz thuh dee-NAH-muh-nay-ter

Rewrite a fraction so no square root remains on the bottom, by multiplying top and bottom by that root. The value stays the same, since you multiply by 1. Like: Rewriting an answer in the standard form the answer key uses.

RAD and DEG mode rad and deeg mohd

The calculator setting that says how to read an angle: RAD reads it as radians, DEG as degrees. A plain number like −7 needs RAD; 30° needs DEG. Like: Choosing miles or kilometers on a car's dashboard.

Vertical asymptote VER-tih-kul AS-im-toht

A vertical line approached by a function graph as its outputs grow without bound in size. Tangent's vertical asymptotes occur at angles it excludes. Like: A vertical fence the curve approaches while its height grows.

Central angle SEN-trul ANG-gul

An angle whose vertex, or pivot, is at the center of a circle. On U its radian measure is the directed arc travel t. Like: A wheel spoke turning around the wheel's center.

Rectangular system rek-TANG-gyuh-lur SIS-tum

The horizontal x-axis and vertical y-axis meet at 90° and locate points by coordinates (x, y). The origin is their intersection. Like: Two straight streets crossing at a right angle.

Sine syne

Sine reads the unit-circle point's height. Like: Height above or below a wheel's center.

Cosine KOH-syne

Cosine reads the unit-circle point's horizontal position. Like: East or west position on a map.

Tangent TAN-junt

Tangent divides height by horizontal position. Like: One ramp direction compared with the other.

Cotangent koh-TAN-junt

Cotangent divides horizontal position by height. Like: One ramp direction compared with the other.

Secant SEE-kunt

Secant is the reciprocal of the horizontal coordinate. Like: Flipping a nonzero measured amount.

Cosecant koh-SEE-kunt

Cosecant is the reciprocal of the height coordinate. Like: Flipping a nonzero measured amount.

Quick checks

Before you start

• Reading the symbols and ordering numbers

  You can read a formula like instructions on a recipe. A letter stands for a number that may change. A real number is a position on the number line, including whole numbers, fractions and decimals. Positive numbers sit to the right of 0; negative numbers sit to the left. The origin is the point marked 0. In this section t and θ, pronounced theta, name angles. α and β, pronounced alpha and beta, name two different angles. A function is a rule that gives one output for each permitted input.

• Adding fractions, distributing and factoring

  You can combine slices only after you cut them to the same size. A fraction's denominator names the size and its numerator counts the pieces. Multiplying the top and bottom by the same nonzero number changes the cuts but preserves the amount. Multiplication can also spread across a sum. Factoring reverses that spreading by collecting a common multiplier.

• Signs when you multiply or divide

  When you multiply or divide two numbers, the sign of the answer depends only on whether their signs match. Matching signs give a positive answer; different signs give a negative answer. A minus sign in front of a negative number makes it positive, like undoing an undo. You will use these rules whenever an output is formed by dividing two signed quantities.

• Coordinates, angles and the unit-circle definitions

  You can locate a spot on a town map by giving two directions. In (x, y), x tells how far right or left and y tells how far up or down. The x-axis is the horizontal line and the y-axis is the vertical line. They meet at the origin (0, 0). A circle's radius is the distance from its center to its rim. The unit circle U has radius 1 and center (0, 0). An angle measures how far a ray turns; the starting ray is its initial side and the stopping ray is its terminal side.

• Function notation: f(−t) and f(t + k)

  f(something) means 'feed that something into the rule'. So f(−t) means feed in the negative of t, and f(t + k) means feed in t plus k. Where the minus sign sits matters: sin(−t) is the sine of the angle −t, with the minus sign inside on the input, while −sin t is the negative of the output. This section is largely about when those two are equal.

• Square roots of fractions, and simplifying roots

  A square root undoes squaring: 9$\sqrt{9}$ = 3 because 3²$3^2$ = 9. Two tools come up in this section. The square root of a fraction is the root of the top over the root of the bottom. And a root can be simplified by pulling out a perfect-square factor: in 8$\sqrt{8}$ = 4 × 2$\sqrt{4\times 2}$, the 4 comes out as its root, 2, and the other 2 stays inside, giving 22$\sqrt{2}$.

• Reciprocals, dividing fractions, rationalizing

  The reciprocal of a nonzero number is 1 divided by it; for a fraction, flip numerator and denominator. Dividing by a nonzero fraction is multiplying by its flip, the way asking how many halves fit in 3 gives 3 × 2 = 6. Rationalizing the denominator means rewriting a fraction with no square root on the bottom. This is a preferred answer convention, not a change in value: 13$\frac{1}{\sqrt{3}}$ and 33$\frac{\sqrt{3}}{3}$ are the same exact number.

• Using a point farther from the center

  You can shrink a map without changing which direction a road points. In the same way, a point on an angle's terminal side can be moved inward to the unit circle without changing the angle. Its coordinates shrink by the same positive factor. Ratios between the coordinates stay the same, while sine and cosine require the coordinates after the radius becomes 1.

• The unit circle and the special values (from 5-2)

  The unit circle is the circle of radius 1 centered at the origin. For an input t, start at (1, 0) and walk a distance t along the circle, counterclockwise if t is positive and clockwise if t is negative. The point where you stop is P(cos t, sin t): its x-coordinate is the cosine and its y-coordinate is the sine. The four axis points and the special angles π6$\frac{\pi }{6}$, π4$\frac{\pi }{4}$, π3$\frac{\pi }{3}$ supply the exact values this section keeps reusing. A right triangle has one 90° angle. Its hypotenuse is its longest side, opposite that angle; relative to the acute angle θ, opposite is the facing leg and adjacent is the touching leg. An acute angle is between 0° and 90°.

• Interval notation and the symbol ∪

  Interval notation is a short way to write a stretch of the number line. A square bracket [ or ] means the endpoint is included; a round parenthesis ( or ) means it is left out. The symbol ∞ always gets a parenthesis, because infinity is not a number you can land on. The symbol ∪, read 'union', glues two stretches together and means 'or'.

• Fractions with π: common denominators

  To add or subtract fractions, the pieces must be the same size, the way you can add quarters to quarters but not quarters to thirds. The symbol π rides along like a unit, the same way 'dollars' rides along in 3 dollars + 2 dollars. So before you compute 11π4$\frac{11\pi }{4}$ − 2π, you rewrite 2π in quarters.

• Mixed numbers like 318$\frac{1}{8}$π

  The notes write some angles as mixed numbers. 318$\frac{1}{8}$π means 3π plus 18$\frac{1}{8}$π, like 3 whole pizzas and one eighth of another. The whole-number part is handy: every 2π is one full turn, so you can drop an even number of π at once. Decimals such as 9.25π or −10.5π work the same way, since 0.25 = 14$\frac{1}{4}$ and 0.5 = 12$\frac{1}{2}$.

• Coterminal angles: same stopping point (from 5-1)

  Two angles are coterminal when they end on the same terminal side. Picture two runners on a circular track who stop at the same spot, one of them after running an extra lap. A full turn is 360°, or 2π in radians, so adding or subtracting full turns never changes where an angle ends. This section uses that idea constantly: it trades a big or negative angle for one between 0 and a full turn. The target includes 0 and excludes the full turn, so an angle ending on the starting side reduces to 0.

• Decimal landmarks for π

  π is a number, about 3.14159, a little more than 3. When an angle is a plain number with no π and no degree sign, like t = −7 or t = 2, you cannot compare fractions of π. Instead you compare decimals with four landmarks, the way you would locate a house number between two cross streets: π2$\frac{\pi }{2}$ ≈ 1.57, π ≈ 3.14, 3π2$\frac{3\pi }{2}$ ≈ 4.71 and 2π ≈ 6.28.

• Solving a squared equation and combining symbolic fractions

  You can think of an equation as a balanced scale. Subtracting the same quantity from each side keeps it balanced. A squared equation tells you a size but can leave two signs possible. A letter can also appear in a fraction. The rules are the same as with number fractions, but you must keep the denominator nonzero.

• Solving a linear equation without losing the balance

  You can find an unknown by undoing operations in reverse order, like unpacking a parcel: remove the outside wrapping before opening the box. An equation stays true if you add, subtract, multiply or divide both sides by the same quantity, provided a divisor is nonzero. A linear equation uses the unknown to the first power, without squaring it.