Unit 3: Trigonometric and Polar Functions

21 terms. Say what each one means, then open the card to check.

• Radian

  The measure of a central angle whose arc is as long as the radius. In general the angle is arc length divided by radius, so a full turn is $2\pi$ radians.

  Example. On a circle of radius 3 cm, an arc 6 cm long sits across a central angle of 2 radians.

• Periodic function

  A function whose outputs repeat over equal intervals of input, so $f(x + p) = f(x)$ for every x and some fixed positive number p.

  Example. A fictional tide gauge that shows the same reading every 12 hours describes a periodic function of time.

• Period

  The smallest positive change in input after which a periodic function repeats. One full cycle of the graph covers an interval of that length.

  Example. $y = \sin x$ has period $2\pi$, and $y = \sin(4x)$ has period $\frac{2\pi}{4} = \frac{\pi}{2}$.

• Unit circle

  The circle of radius 1 centered at the origin. The terminal ray of an angle $\theta$ in standard position meets it at the point $(\cos\theta, \sin\theta)$.

  Example. The angle $\frac{\pi}{3}$ meets the unit circle at $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$.

• Reference angle

  The acute angle between an angle's terminal ray and the x-axis. Trig values of the angle equal those of its reference angle, up to a sign set by the quadrant.

  Example. $\frac{5\pi}{6}$ has reference angle $\frac{\pi}{6}$, so $\sin\frac{5\pi}{6} = \frac{1}{2}$, positive in the second quadrant.

• Tangent

  The ratio $\tan\theta = \frac{\sin\theta}{\cos\theta}$, which is the slope of the terminal ray. It is undefined where $\cos\theta = 0$, and $y = \tan x$ has period $\pi$.

  Example. $\tan\frac{\pi}{4} = 1$ because the ray at $\frac{\pi}{4}$ rises 1 unit for every 1 unit it runs.

• Sinusoidal function

  A function of the form $a\sin(b(x - c)) + d$ or $a\cos(b(x - c)) + d$: a sine or cosine graph that has been stretched, shifted or both.

  Example. $y = 2\cos(3x) + 5$ is sinusoidal, and its outputs wave between 3 and 7.

• Amplitude

  Half the distance between the maximum and minimum of a sinusoid, equal to $|a|$. It is a distance, so it is never negative.

  Example. A tide that runs from 2 feet to 10 feet has amplitude $\frac{10 - 2}{2} = 4$ feet.

• Midline

  The horizontal line $y = d$ halfway between a sinusoid's maximum and minimum. The graph crosses it twice in each period.

  Example. For $y = -3\sin x + 8$ the midline is $y = 8$, halfway between the minimum 5 and the maximum 11.

• Phase shift

  The horizontal shift c in $a\sin(b(x - c)) + d$. Factor b out of the inside first, or the shift you read will be wrong.

  Example. $y = \sin(2x - \pi) = \sin\left(2\left(x - \frac{\pi}{2}\right)\right)$ is the graph of $\sin(2x)$ shifted $\frac{\pi}{2}$ to the right.

• Inverse sine

  The function $\arcsin x$, also written $\sin^{-1} x$, which gives the angle in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ whose sine is x. Its input must be between -1 and 1.

  Example. $\arcsin\left(-\frac{1}{2}\right) = -\frac{\pi}{6}$, not $\frac{7\pi}{6}$, because the output is restricted.

• Restricted domain

  A piece of a function's domain on which it is one-to-one, chosen so the function has an inverse. Sine uses $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ and cosine uses $[0, \pi]$.

  Example. Cosine on $[0, \pi]$ takes each value from 1 down to -1 exactly once, so arccos can undo it there.

• Solutions of a trig equation

  The angles that make the equation true. For $\sin x = k$ with $-1 < k < 1$ there are two in each interval of length $2\pi$; adding multiples of $2\pi$ gives the rest.

  Example. $\cos x = \frac{1}{2}$ has the solutions $\frac{\pi}{3}$ and $\frac{5\pi}{3}$ in the interval $[0, 2\pi)$.

• Pythagorean identity

  $\sin^2\theta + \cos^2\theta = 1$ for every angle, from the unit circle. Dividing by $\cos^2\theta$ gives $1 + \tan^2\theta = \sec^2\theta$.

  Example. If $\sin\theta = 0.6$, then $\cos^2\theta = 1 - 0.36 = 0.64$, so $\cos\theta$ is 0.8 or -0.8.

• Secant, cosecant and cotangent

  The reciprocals $\sec\theta = \frac{1}{\cos\theta}$, $\csc\theta = \frac{1}{\sin\theta}$ and $\cot\theta = \frac{1}{\tan\theta}$. Each is undefined where its denominator is 0.

  Example. $\cos\frac{\pi}{3} = \frac{1}{2}$, so $\sec\frac{\pi}{3} = 2$.

• Sum identities

  $\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$ and $\cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta$. They give exact values for sums of special angles.

  Example. $\sin\frac{7\pi}{12} = \sin\left(\frac{\pi}{3} + \frac{\pi}{4}\right) = \frac{\sqrt{6} + \sqrt{2}}{4}$.

• Double-angle identities

  $\sin 2\theta = 2\sin\theta\cos\theta$ and $\cos 2\theta = \cos^2\theta - \sin^2\theta$, which also equals $1 - 2\sin^2\theta$ and $2\cos^2\theta - 1$.

  Example. If $\sin\theta = \frac{3}{5}$ and $\cos\theta = \frac{4}{5}$, then $\sin 2\theta = 2 \cdot \frac{3}{5} \cdot \frac{4}{5} = \frac{24}{25}$.

• Polar coordinates

  A point named $(r, \theta)$: a directed distance r from the origin along the ray at angle $\theta$. Then $x = r\cos\theta$, $y = r\sin\theta$ and $r^2 = x^2 + y^2$.

  Example. The polar point $\left(2, \frac{\pi}{2}\right)$ is the rectangular point $(0, 2)$.

• Polar function

  A rule $r = f(\theta)$ that gives a directed distance for each angle. Its graph is traced as $\theta$ sweeps around the origin.

  Example. $r = 3$ is a circle of radius 3 centered at the origin, and $r = \theta$ for $\theta \ge 0$ is a spiral.

• Negative r

  When r is negative, the point is plotted |r| units from the origin in the direction opposite the ray at angle $\theta$, that is, along the ray at $\theta + \pi$.

  Example. The polar point $\left(-2, \frac{\pi}{2}\right)$ is 2 units straight down, at the rectangular point $(0, -2)$.

• Distance from the origin

  On a polar graph the distance from the origin is |r|. If r is positive and increasing, or negative and decreasing, the point moves away from the origin.

  Example. For $r = \theta - 2$ on $0 < \theta < 2$, r is negative and increasing, so the point moves toward the origin.