Unit 3

Trigonometric and Polar Functions: the one-page sheet

3.1 Periodic Phenomena

A periodic function repeats its outputs every P units of input, f(x + P) = f(x) , and the smallest such P is the period. Learn one cycle and you know the function everywhere: shift any input by whole periods into the cycle. Increasing and decreasing intervals, concavity and rates of change repeat every period, and the average rate of change over a full period is 0 .

A periodic function repeats its outputs: there is a positive number $P$ with $f(x + P) = f(x)$ for every input $x$ . The smallest such $P$ is the period.
One full cycle, any interval of length $P$ , holds all the information. Shift inputs by whole periods to reach any other input.
Over any interval one full period long, the average rate of change is $0$ , because the output ends where it started.
Two inputs can share an output without being a period apart: a cycle usually passes through each in-between value twice.

f is periodic, and the smallest such positive P is the period: f(x + P) = f(x) for every x copy that piece of graph left and right, P units at a time, to get the whole graph add or subtract whole periods until the input lands in the cycle: with P = 6 , f(47) = f(47 − 7⋅6) = f(5) intervals where f increases or decreases, concavity and average rates of change all repeat every P units

periodic function: A function whose outputs repeat at regular intervals of input: there is a positive number $P$ with $f(x + P) = f(x)$ for every $x$ . The smallest such $P$ is the period, the length of one full cycle.

3.2 Sine, Cosine, and Tangent

An angle in radians is arc length over radius, so a full turn is 2π . Place the angle θ in standard position; its terminal ray meets the unit circle at (cos θ, sin θ) , and tan θ = (sin θ)/(cos θ) is the ray's slope. These definitions work for every angle, and the quadrant sets the signs.

Radians: $\theta = \frac{\text{arc length}}{\text{radius}}$ . On the unit circle, an angle of $\theta$ radians cuts off an arc of length exactly $\theta$ .
Convert with $\pi$ radians $= 180^\circ$ : multiply degrees by $\frac{\pi}{180}$ , or radians by $\frac{180}{\pi}$ .
Signs follow the quadrant: in Quadrant II, $\cos\theta < 0$ and $\sin\theta > 0$ ; in Quadrant III both are negative; in Quadrant IV, $\cos\theta > 0$ and $\sin\theta < 0$ .
Angles that differ by $2\pi$ share a terminal ray, so they have the same sine, cosine and tangent.

that is its radian measure: a full turn is 2πr/r = 2π radians, so π radians = 180^(∘) its terminal ray meets the unit circle x² + y² = 1 at one point its x -coordinate is cos θ and its y -coordinate is sin θ , for any angle, not only acute ones tan θ = (sin θ)/(cos θ) , the slope of the terminal ray, undefined where cos θ = 0

radian: A unit of angle measure: an angle's radian measure is the length of the arc it cuts off divided by the radius. A full turn is $2\pi$ radians, so $\pi$ radians $= 180^\circ$ . On the unit circle, an angle of $\theta$ radians cuts off an arc of length $\theta$ .
unit circle: The circle of radius 1 centered at the origin, $x^2 + y^2 = 1$ . The terminal ray of an angle $\theta$ in standard position meets it at the point $(\cos\theta, \sin\theta)$ .
standard position: An angle placed with its vertex at the origin and its initial ray along the positive $x$ -axis. It turns to its terminal ray, counterclockwise for a positive angle and clockwise for a negative one.
sine function: The sine and cosine functions, and with them tangent, defined for any angle $\theta$ in standard position: $\cos\theta$ and $\sin\theta$ are the $x$ - and $y$ -coordinates of the point where the terminal ray meets the unit circle, and $\tan\theta = \frac{\sin\theta}{\cos\theta}$ is the slope of the terminal ray.

3.3 Sine and Cosine Function Values

Exact sines and cosines at multiples of π/6 and π/4 come from the 30-60-90 and 45-45-90 triangles. Find the reference angle, use its first-quadrant value, and give it the sign of the quadrant. On a circle of radius r , the point at angle θ is (rcos θ, rsin θ) .

The special angles are the multiples of $\frac{\pi}{6}$ and $\frac{\pi}{4}$ . Their sines and cosines are $0$ , $\pm\frac{1}{2}$ , $\pm\frac{\sqrt{2}}{2}$ , $\pm\frac{\sqrt{3}}{2}$ or $\pm 1$ .
Method: find the reference angle, take the first-quadrant value, then give it the sign of the quadrant.
For a reference angle of $\frac{\pi}{6}$ or $\frac{\pi}{3}$ , the coordinate that is larger in size is $\frac{\sqrt{3}}{2}$ : it belongs to the axis the ray is closer to.
On a circle of radius $r$ centered at the origin, the point at angle $\theta$ is $(r\cos\theta, r\sin\theta)$ .

it is a special right triangle, so the coordinates are exact: ((√3)/2 , 1/2) , ((√2)/2 , (√2)/2) , (1/2 , (√3)/2) its reference angle, the acute angle between the terminal ray and the x -axis, is π/6 , so its coordinates have the same sizes 5π/6 is in Quadrant II, so cos 5π/6 = −(√3)/2 and sin 5π/6 = 1/2 scale the coordinates by r : the point at angle θ is (rcos θ, rsin θ)

special angle: A multiple of $\frac{\pi}{6}$ or $\frac{\pi}{4}$ , whose sine and cosine are known exactly from the special right triangles. Its reference angle, the acute angle between its terminal ray and the $x$ -axis, gives the size of each value; the quadrant gives the sign.

3.4 Sine and Cosine Function Graphs

The graph of y = sin θ is the height of the point at angle θ on the unit circle; y = cos θ is its horizontal position. Both repeat every 2π and stay between −1 and 1 , and cosine is sine shifted π/2 to the left. Each graph is concave down above the axis and concave up below it, steepest at its zeros and flat at its highest and lowest points.

Both graphs have domain all real numbers, range $[-1, 1]$ and period $2\pi$ .
Five key points per cycle: sine $(0, 0),$ $\left(\frac{\pi}{2}, 1\right),$ $(\pi, 0),$ $\left(\frac{3\pi}{2}, -1\right),$ $(2\pi, 0)$ ; cosine $(0, 1),$ $\left(\frac{\pi}{2}, 0\right),$ $(\pi, -1),$ $\left(\frac{3\pi}{2}, 0\right),$ $(2\pi, 1)$ .
Sine is zero at every multiple of $\pi$ ; cosine is zero at $\frac{\pi}{2}$ plus every multiple of $\pi$ .
Each graph is concave down above the $\theta$ -axis and concave up below it, so its rate of change is decreasing where the graph is above the axis.

you get y = sin θ : 0 at θ = 0 , up to 1 at π/2 , back to 0 at π , down to −1 at 3π/2 , back to 0 at 2π you get y = cos θ : 1, 0, −1, 0, 1 at the same five inputs, the sine graph moved π/2 to the left both graphs repeat every 2π and stay between −1 and 1 the rate of change of sin θ is largest in size at the zeros, and the graph is concave down where sin θ > 0 and concave up where sin θ < 0

sine curve: The graph of $y = \sin\theta$ or $y = \cos\theta$ , traced from the unit circle: sine is the point's height and cosine its horizontal position. Both repeat every $2\pi$ and oscillate between $-1$ and $1$ ; the cosine curve is the sine curve shifted $\frac{\pi}{2}$ to the left.

3.5 Sinusoidal Functions

A sinusoidal function is a sine or cosine graph after dilations and translations. Four numbers describe it: the midline y = (max + min)/2 , the amplitude (max − min)/2 , the period (the length of one cycle) and the frequency, one over the period. Sine is odd and cosine is even.

A sinusoidal function has the shape of a sine graph after dilations and translations. $y = \sin\theta$ and $y = \cos\theta$ are both sinusoidal, each with amplitude 1, midline $y = 0$ and period $2\pi$ .
Amplitude is a distance, so it is never negative. Midline is a line, so write it as an equation: $y = 5$ .
Frequency $= \frac{1}{\text{period}}$ . A period of 8 seconds means $\frac{1}{8}$ of a cycle per second; 220 cycles per second means a period of $\frac{1}{220}$ second.
Sine is odd, $\sin(-\theta) = -\sin\theta$ ; cosine is even, $\cos(-\theta) = \cos\theta$ .

you get a sinusoidal function: the same smooth wave shape, with new heights and a new cycle length the midline y = (max + min)/2 , the horizontal line the wave oscillates around the amplitude (max − min)/2 , how far the wave reaches above and below its midline the period P ; the frequency 1/P counts the cycles per unit of input

sinusoidal function: A function whose graph is the graph of sine (or cosine) after vertical and horizontal dilations and translations: a smooth wave that oscillates evenly about a horizontal line. Described by its amplitude, midline, period and frequency.
amplitude: Half the distance between the greatest and least values of a sinusoidal function, $\frac{\max - \min}{2}$ : how far the graph reaches above and below its midline. It is never negative.
midline: The horizontal line halfway between the greatest and least values of a sinusoidal function, $y = \frac{\max + \min}{2}$ . The graph oscillates around it.
frequency: The number of cycles a periodic function completes per unit of input: the reciprocal of the period. A period of 4 seconds is a frequency of $\frac{1}{4}$ cycle per second.

3.6 Sinusoidal Function Transformations

In f(θ) = asin (b(θ + c)) + d , |a| is the amplitude, 2π/(|b|) the period, c the phase shift (left by c ) and y = d the midline; cosine works the same way. To build a formula from a graph, read d and |a| , get b = 2π/period , then find c from a landmark point. Factor out b before reading a shift.

For $f(\theta)$ $= a\sin(b(\theta + c)) + d$ or $a\cos(b(\theta + c)) + d$ : amplitude $|a|$ , period $\frac{2\pi}{|b|}$ , phase shift $-c$ (left by $c$ ), midline $y = d$ .
Factor $b$ out before reading the shift: $\sin(2\theta + \pi)$ $= \sin\left(2\left(\theta + \frac{\pi}{2}\right)\right)$ moves left $\frac{\pi}{2}$ , not $\pi$ .
Building from a graph: $d$ from the midline, $|a|$ from the amplitude, $b = \frac{2\pi}{\text{period}}$ , then $c$ from one landmark (a rising midline crossing for sine, a highest point for cosine).
Many formulas give the same graph: $\cos\theta = \sin\left(\theta + \frac{\pi}{2}\right)$ , and $c$ can change by any whole number of periods.

a vertical dilation: the amplitude becomes |a| , and a negative a also reflects the graph across the midline a horizontal dilation: one cycle now takes 2π/(|b|) , so the period is 2π/(|b|) and b = 2π/period a phase shift: the graph moves left c units if c > 0 , right if c < 0 a vertical translation: the midline becomes y = d

phase shift: The horizontal translation of a sinusoidal function. In $f(\theta)$ $= a\sin(b(\theta + c)) + d$ the graph of $a\sin(b\theta) + d$ moves left $c$ units when $c > 0$ and right $|c|$ units when $c < 0$ . Factor $b$ out of the input before reading it.

3.7 Sinusoidal Function Context and Data Modeling

To model a periodic context, get the midline and amplitude from the greatest and least values, the period (and b = 2π/period ) from the cycle length or frequency, and the shift from one landmark time. With data, sinusoidal regression does the same job. Use the model on the domain the context supports, and describe change on an interval as two statements: the direction of the quantity and the direction of its rate of change.

Five labeled points on one cycle are: a highest or lowest point, a midline crossing, the opposite extreme, the other midline crossing and the first extreme again, each a quarter period apart.
Easy landmarks: highest at $t_0$ gives $a\cos(b(t - t_0)) + d$ with $a > 0$ ; lowest at $t_0$ gives the same with $a < 0$ ; midline and rising at $t_0$ gives $a\sin(b(t - t_0)) + d$ with $a > 0$ .
Describe change on an interval in two separate statements: whether the quantity is increasing or decreasing, and whether its rate of change is increasing or decreasing (concave up or concave down).
A model fits the facts it was built from. Past the data, or if the real cycle drifts, predictions get worse.

the midline d = (max + min)/2 and the amplitude |a| = (max − min)/2 the period ( 1/frequency if needed), then b = 2π/period choose sine or cosine and the sign of a to match that landmark, then set c so the landmark lands at the right time it is trustworthy only on the domain the context supports; with data, a sinusoidal regression on a calculator builds the same kind of model

sinusoidal model: A sinusoidal function $a\sin(b(t + c)) + d$ or $a\cos(b(t + c)) + d$ used to describe a periodic context. Its constants come from the greatest and least values, the period and one landmark time, or from a sinusoidal regression on data. It is valid on the domain the context supports.

3.8 The Tangent Function

tan θ = (sin θ)/(cos θ) , the slope of the terminal ray. It is 0 at θ = kπ , has vertical asymptotes at θ = π/2 + kπ with outputs growing without bound on each side, and has period π . For atan (b(x − c)) + d the period is π/(|b|) and the asymptotes shift with the graph.

$\tan\theta$ is the slope of the terminal ray, and the height where that ray meets the line $x = 1$ .
Tangent is $0$ where sine is $0$ and has a vertical asymptote where cosine is $0$ .
The period of $a\tan(bx)$ is $\frac{\pi}{|b|}$ , half of what the sine rule would give.
On each branch tangent increases; it is concave down left of its zero and concave up right of it.

tangent is 0 where sin θ = 0 , at θ = kπ the quotient is undefined there: vertical asymptotes their quotient repeats, so the period is π period π/(|b|) , shifted right c and up d , stretched by a

tangent function: The function $\tan x = \frac{\sin x}{\cos x}$ : the slope of the terminal ray of the angle $x$ . Its period is $\pi$ , its zeros are at $x = k\pi$ , and it has vertical asymptotes at $x = \frac{\pi}{2} + k\pi$ , where cosine is $0$ .

3.9 Inverse Trigonometric Functions

Restricted to [−π/2 , π/2] , [0, π] and (−π/2 , π/2) , sine, cosine and tangent are invertible. sin⁻¹ , cos⁻¹ and tan⁻¹ take a ratio and return the one angle in that interval. Inverse is not reciprocal, and sin⁻¹ (sin x) = x only for x in the restricted interval.

An inverse trigonometric function takes a ratio and returns an angle.
$\sin^{-1}$ and $\tan^{-1}$ return angles between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$ ; $\cos^{-1}$ returns angles between $0$ and $\pi$ .
$\sin^{-1} x$ is the inverse function, not the reciprocal $\frac{1}{\sin x}$ .
For a composite like $\cos^{-1}(\cos x)$ , find the inner value first, then the angle in the right range.

they have no inverse on their whole domains sin on [−π/2 , π/2] , cos on [0, π] , tan on (−π/2 , π/2) sin⁻¹ , cos⁻¹ and tan⁻¹ , whose outputs are angles in those intervals you get x back only when x is already in [−π/2 , π/2]

inverse sine: The inverse sine, inverse cosine and inverse tangent functions, $\sin^{-1} x$ , $\cos^{-1} x$ and $\tan^{-1} x$ (also written $\arcsin x$ , $\arccos x$ , $\arctan x$ ). Each takes a ratio and returns an angle: $\sin^{-1}$ and $\tan^{-1}$ in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ (open for tangent), $\cos^{-1}$ in $[0, \pi]$ . They are inverses, not reciprocals.

3.10 Trigonometric Equations and Inequalities

To solve a trigonometric equation, isolate the function, take the principal value with the inverse, find the second angle by symmetry, then add periods until you have every solution in the interval and none outside it. Inequalities follow the equation: its solutions cut the interval into pieces, and a test value decides each piece.

Sine and cosine take most values twice per period; the inverse key gives only one of them.
Second angle: $\pi - \theta$ for sine, $2\pi - \theta$ (or $-\theta$ ) for cosine, $\theta + \pi$ for tangent.
For $\sin(bx) = k$ , solve for $bx$ on the stretched interval first, then divide by $b$ .
For an inequality, solve the equation, then test one value in each piece.

an equation like sin x = 1/2 one principal value, such as π/6 the second angle in the same period every solution in the interval, and no extras

trigonometric equation: An equation or inequality in which the unknown is the input of a trigonometric function, such as $2\sin x = 1$ or $\cos(2x) \le 0$ . It usually has two solutions per period; to find all of them in an interval, isolate the function, take the principal value, use symmetry for the second angle, and add periods.

3.11 The Secant, Cosecant, and Cotangent Functions

sec x = 1/(cos x) , csc x = 1/(sin x) and cot x = (cos x)/(sin x) . Each has vertical asymptotes where its denominator is 0 . Secant and cosecant equal their partners at ±1 and never take values strictly between −1 and 1 ; cotangent takes every real value and has period π .

Pair them by the letter they do not share: $\sec$ goes with $\cos$ , $\csc$ with $\sin$ .
$\sec$ has asymptotes where $\cos x = 0$ ; $\csc$ and $\cot$ where $\sin x = 0$ .
Where $\cos x$ decreases and stays positive, $\sec x$ increases: reciprocals move the opposite way.
$\sec$ and $\csc$ have period $2\pi$ and range $(-\infty, -1] \cup [1, \infty)$ ; $\cot$ has period $\pi$ and range $(-\infty, \infty)$ .

sec x = 1/(cos x) , csc x = 1/(sin x) , cot x = (cos x)/(sin x) a vertical asymptote: sec where cos x = 0 , csc and cot where sin x = 0 the reciprocal equals it there: a turning point of sec or csc |sec x| ≥ 1 and |csc x| ≥ 1 , so neither takes values strictly between −1 and 1

secant function: The secant, cosecant and cotangent functions: $\sec x = \frac{1}{\cos x}$ , $\csc x = \frac{1}{\sin x}$ and $\cot x = \frac{\cos x}{\sin x}$ . Each has vertical asymptotes where its denominator is $0$ . Secant and cosecant have range $(-\infty, -1] \cup [1, \infty)$ and period $2\pi$ ; cotangent has range $(-\infty, \infty)$ and period $\pi$ .

3.12 Equivalent Representations of Trigonometric Functions

The Pythagorean identity sin² x + cos² x = 1 and its forms with tan , sec , cot and csc ; the sum and difference identities for sine and cosine; and the double-angle identities sin (2x) = 2sin xcos x and cos (2x) = cos² x − sin² x = 2cos² x − 1 = 1 − 2sin² x . Use them to find exact values, to rewrite expressions as one function, and to solve equations.

An identity is true for every input where both sides are defined; you use it to swap one form for another.
$\sin(a + b)$ is not $\sin a + \sin b$ ; use the sum identity.
$\cos(2x)$ has three forms: $\cos^2 x - \sin^2 x$ , $2\cos^2 x - 1$ and $1 - 2\sin^2 x$ . Pick the one that matches the rest of the expression.
To rewrite as a single trigonometric function, write everything in sines and cosines, simplify, and stop at one function.

sin² x + cos² x = 1 , the Pythagorean identity tan² x + 1 = sec² x and 1 + cot² x = csc² x sin (a + b) = sin acos b +cos asin b and cos (a + b) = cos acos b − sin asin b sin (2a) = 2sin acos a and cos (2a) = cos² a − sin² a

Pythagorean identity: The identity $\sin^2 x + \cos^2 x = 1$ , true because $(\cos x, \sin x)$ is on the unit circle, and its forms $\tan^2 x + 1 = \sec^2 x$ and $1 + \cot^2 x = \csc^2 x$ .
sum identity: The sum and difference identities $\sin(a \pm b)$ $= \sin a\cos b \pm \cos a\sin b$ and $\cos(a \pm b)$ $= \cos a\cos b \mp \sin a\sin b$ , and the double-angle identities $\sin(2a) = 2\sin a\cos a$ and $\cos(2a)$ $= \cos^2 a - \sin^2 a$ $= 2\cos^2 a - 1$ $= 1 - 2\sin^2 a$ that follow from them.
trigonometric identity: An equation with trigonometric functions that is true for every input where both sides are defined, such as $\sin^2 x + \cos^2 x = 1$ . Identities let you replace an expression with an equivalent one to find exact values, simplify, or solve equations.

3.13 Trigonometry and Polar Coordinates

A point in polar coordinates (r, θ) is r units from the pole in the direction θ . Convert with x = rcos θ , y = rsin θ , r = √(x² + y²) and tan θ = y/x , checking the quadrant. Every point has many polar names, and a complex number a + bi can be written as r(cos θ + isin θ) .

$(r, \theta)$ means: face the direction $\theta$ , then move $r$ units. A negative $r$ moves backward through the pole.
$x = r\cos\theta$ , $y = r\sin\theta$ and $r^2 = x^2 + y^2$ convert between the two systems.
$\tan^{-1}\frac{y}{x}$ gives an angle in Quadrant I or IV; if the point is in Quadrant II or III, add $\pi$ .
A complex number $a + bi$ is the point $(a, b)$ , so it has a polar form $r(\cos\theta + i\sin\theta)$ .

the point (r, θ) : turn to θ , then go r units out from the pole x = rcos θ and y = rsin θ r = √(x² + y²) and tan θ = y/x , with the quadrant checked the same point gets another name

polar coordinates: A way to name a point by its distance $r$ from the pole (the origin) and the angle $\theta$ from the polar axis (the positive $x$ -axis): $(r, \theta)$ . They convert with $x = r\cos\theta$ , $y = r\sin\theta$ and $r^2 = x^2 + y^2$ . A negative $r$ points the opposite way, and each point has many polar names.
polar form of a complex number: The form $r(\cos\theta + i\sin\theta)$ of a complex number $a + bi$ , from the polar coordinates of the point $(a, b)$ in the complex plane: $r = \sqrt{a^2 + b^2}$ is its distance from $0$ and $\theta$ its angle.

3.14 Polar Function Graphs

A polar function r = f(θ) assigns a signed distance to every direction. Read it from its graph of r against θ : zeros are visits to the pole, negative values are drawn on the opposite ray, and each hump draws a petal or loop. Circles ( r = a , r = acos θ ), limaçons and cardioids ( r = a + bcos θ ) and roses ( r = asin (nθ) ) are the families to know.

A polar function $r = f(\theta)$ gives a distance for each direction; the curve is all points $(f(\theta), \theta)$ .
Each hump of the graph of $r$ against $\theta$ draws one loop or petal; zeros of $r$ are visits to the pole.
$r = a\cos(n\theta)$ and $r = a\sin(n\theta)$ are roses: $n$ petals if $n$ is odd, $2n$ if $n$ is even.
$r = a + b\cos\theta$ is a limaçon: an inner loop if $|a| < |b|$ , a cardioid if $|a| = |b|$ , no loop if $|a| > |b|$ .

the input of r = f(θ) is the direction the output is the signed distance along that direction the curve passes through the pole the point is drawn on the opposite ray, in the opposite quadrant

polar function: A function $r = f(\theta)$ whose input is an angle and whose output is a signed distance from the pole. Its graph in the polar plane is all points $(f(\theta), \theta)$ ; families include circles, limaçons and cardioids ( $r = a + b\cos\theta$ ) and rose curves ( $r = a\sin(n\theta)$ , $r = a\cos(n\theta)$ ).

3.15 Rates of Change in Polar Functions

For a polar function, the average rate of change of r on [a, b] is (r(b) − r(a))/(b − a) units per radian, the slope of a secant line on the graph of r against θ . The distance from the origin is |r| : it increases where r is positive and increasing or negative and decreasing, and decreases in the other two cases. Extremes of |r| give the farthest and closest points.

The average rate of change of $r$ on $[a, b]$ is $\frac{r(b) - r(a)}{b - a}$ : the slope of the secant line on the graph of $r$ against $\theta$ .
Whether the curve moves toward or away from the origin depends on the sign of $r$ and on whether $r$ is increasing.
Positive and increasing, or negative and decreasing: moving away. The other two cases: moving closer.
The largest $|r|$ is the farthest point; $r = 0$ is a visit to the origin.

(r(b) − r(a))/(b − a) is the average rate of change of r , in units per radian it increases where r > 0 and increasing, or r < 0 and decreasing the distance from the origin decreases a point of the curve farthest from the origin nearby

rate of change of r: How $r$ changes as $\theta$ changes for a polar function. The average rate of change of $r$ on $[a, b]$ is $\frac{r(b) - r(a)}{b - a}$ , in units per radian. The distance from the origin, $|r|$ , increases where $r$ is positive and increasing or negative and decreasing, and decreases otherwise.