Topic 3: Trigonometric and Polar Functions

Many real phenomena repeat: tides, daylight hours, sound waves, a Ferris wheel rider's height, alternating current. Linear, polynomial, and exponential models cannot capture this rhythmic rise and fall, so AP Precalculus introduces the trigonometric functions, whose defining feature is periodicity. This unit begins with the unit circle, which turns angles into coordinates and coordinates into the values of sine and cosine. From there you learn to read and reshape sinusoidal graphs using amplitude, period, midline, and phase shift, then run the process in reverse to model data. The unit closes by introducing inverse trig functions, trig equation solving, the foundational identities, and an entirely new way to locate points: polar coordinates. Mastering the connections among angle, coordinate, graph, and equation is the central goal.

Radians and the Unit Circle

A radian measures an angle by the arc length it subtends on a circle of radius 1. Because a full circle has circumference $2\\pi r$, one full revolution is $2\\pi$ radians, so $360^\\circ = 2\\pi$ radians and $180^\\circ = \\pi$ radians. To convert, multiply degrees by $\\frac{\\pi}{180}$, or multiply radians by $\\frac{180}{\\pi}$. The unit circle is the circle $x^2 + y^2 = 1$ centered at the origin. For an angle $\\theta$ measured counterclockwise from the positive $x$-axis, the terminal point on the unit circle is exactly $(\\cos\\theta, \\sin\\theta)$. This is the master definition: cosine is the $x$-coordinate and sine is the $y$-coordinate. Key reference angles you should know cold include $\\theta = \\frac{\\pi}{6}, \\frac{\\pi}{4}, \\frac{\\pi}{3}, \\frac{\\pi}{2}$, giving points such as $\\left(\\frac{\\sqrt{3}}{2}, \\frac{1}{2}\\right)$ at $\\frac{\\pi}{6}$ and $\\left(\\frac{\\sqrt{2}}{2}, \\frac{\\sqrt{2}}{2}\\right)$ at $\\frac{\\pi}{4}$. Signs of the coordinates follow the quadrant: in Quadrant II cosine is negative and sine positive, and so on. Arc length on a circle of radius $r$ is $s = r\\theta$ when $\\theta$ is in radians, a clean formula that explains why radians are the natural unit for circular motion.

The Sine, Cosine, and Tangent Functions

Reading the unit circle as $\\theta$ increases generates the three core functions. The parent sine function $y = \\sin\\theta$ starts at $0$, rises to $1$ at $\\frac{\\pi}{2}$, returns to $0$ at $\\pi$, falls to $-1$ at $\\frac{3\\pi}{2}$, and returns to $0$ at $2\\pi$. The parent cosine $y = \\cos\\theta$ is the same wave shifted: it starts at $1$, drops to $0$ at $\\frac{\\pi}{2}$, reaches $-1$ at $\\pi$, and back to $1$ at $2\\pi$. Both have domain all real numbers, range $[-1, 1]$, and period $2\\pi$. Sine is odd, meaning $\\sin(-\\theta) = -\\sin\\theta$, while cosine is even, $\\cos(-\\theta) = \\cos\\theta$. Tangent is defined as $\\tan\\theta = \\frac{\\sin\\theta}{\\cos\\theta}$, which is the slope of the terminal ray. It is undefined wherever $\\cos\\theta = 0$, producing vertical asymptotes at $\\theta = \\frac{\\pi}{2} + k\\pi$ for integer $k$. Tangent has period $\\pi$, not $2\\pi$, range all real numbers, and is odd. The reciprocal functions secant, cosecant, and cotangent are $\\frac{1}{\\cos\\theta}$, $\\frac{1}{\\sin\\theta}$, and $\\frac{1}{\\tan\\theta}$ respectively. A useful structural idea: sinusoids change concavity at their midline crossings and reach extrema where the rate of change is momentarily zero.

Amplitude, Period, Midline, and Phase Shift

A general sinusoid is written $y = a\\sin(b(x - c)) + d$ or with cosine in place of sine. Each constant controls one geometric feature. The amplitude is $|a|$, the distance from the midline to a peak; it is always taken as a magnitude, so a negative $a$ also reflects the graph vertically. The midline is the horizontal line $y = d$, the average of the maximum and minimum values, so $d = \\frac{\\text{max} + \\text{min}}{2}$ and amplitude $= \\frac{\\text{max} - \\text{min}}{2}$. The value $b$ controls horizontal stretch through the period, given by $\\text{period} = \\frac{2\\pi}{|b|}$ for sine and cosine. A larger $|b|$ compresses the wave so it cycles faster. The phase shift is $c$, the horizontal translation: $y = \\sin(b(x - c))$ shifts the parent right by $c$ when $c \\gt 0$. Be careful to factor out $b$ before reading the phase shift, since $y = \\sin(2x - \\pi) = \\sin\\left(2\\left(x - \\frac{\\pi}{2}\\right)\\right)$ shifts right by $\\frac{\\pi}{2}$, not by $\\pi$. For example, $y = 3\\cos\\left(2\\left(x - \\frac{\\pi}{4}\\right)\\right) + 5$ has amplitude $3$, period $\\pi$, midline $y = 5$, and a rightward shift of $\\frac{\\pi}{4}$, so it oscillates between $2$ and $8$.

Sinusoidal Modeling

Modeling reverses the analysis: given a periodic situation, you build the equation. Start by identifying the maximum and minimum of the quantity to get amplitude and midline. Next find the period from how long one full cycle takes, then solve $b = \\frac{2\\pi}{\\text{period}}$. Finally choose phase shift to align the starting behavior, often picking cosine when the cycle begins at a maximum or minimum and sine when it begins at the midline moving upward. Consider a Ferris wheel of diameter $40$ meters whose lowest point is $2$ meters above the ground, completing one revolution every $60$ seconds, with a rider starting at the bottom. The amplitude is the radius $20$, the midline is $2 + 20 = 22$, and the period $60$ gives $b = \\frac{2\\pi}{60} = \\frac{\\pi}{30}$. Because the rider starts at the minimum, a negative cosine fits naturally: $h(t) = -20\\cos\\left(\\frac{\\pi}{30}t\\right) + 22$, where $t$ is seconds and $h$ is meters. Always state the domain restriction implied by context and check units. When modeling temperature over a year or daylight over months, the same recipe applies: extract amplitude, midline, period, and shift, then verify the model reproduces a known data point.

Inverse Trig Functions and Solving Trig Equations

Because sine, cosine, and tangent are periodic, they fail the horizontal line test and are not one-to-one over their full domains. To define inverses, each is restricted to an interval on which it is monotonic. The inverse sine, $\\arcsin x$, has domain $[-1, 1]$ and range $\\left[-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right]$. The inverse cosine, $\\arccos x$, has domain $[-1, 1]$ and range $[0, \\pi]$. The inverse tangent, $\\arctan x$, has domain all reals and range $\\left(-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right)$. An inverse trig function returns only the single value inside its restricted range, called the principal value. Solving a trig equation usually demands more than the principal value because there are infinitely many solutions. To solve $2\\sin x = 1$, first get $\\sin x = \\frac{1}{2}$; the principal solution is $x = \\frac{\\pi}{6}$, but sine is also positive in Quadrant II at $x = \\frac{5\\pi}{6}$, and the full solution set is $x = \\frac{\\pi}{6} + 2\\pi k$ and $x = \\frac{5\\pi}{6} + 2\\pi k$ for integer $k$. For tangent equations use period $\\pi$: $\\tan x = 1$ gives $x = \\frac{\\pi}{4} + \\pi k$. Always use the unit circle to find every angle in one period, then add the period times an integer to capture all solutions.

Trigonometric Identities

Identities are equations true for every value where both sides are defined, and they let you rewrite expressions to simplify or solve. The foundational one is the Pythagorean identity $\\sin^2\\theta + \\cos^2\\theta = 1$, which follows directly from $x^2 + y^2 = 1$ on the unit circle. Dividing it by $\\cos^2\\theta$ yields $1 + \\tan^2\\theta = \\sec^2\\theta$, and dividing by $\\sin^2\\theta$ yields $1 + \\cot^2\\theta = \\csc^2\\theta$. The quotient identity $\\tan\\theta = \\frac{\\sin\\theta}{\\cos\\theta}$ and the reciprocal identities connect the six functions. The even-odd identities, $\\cos(-\\theta) = \\cos\\theta$ and $\\sin(-\\theta) = -\\sin\\theta$, reflect the symmetry of the parent graphs. The sum and difference formulas, such as $\\sin(\\alpha + \\beta) = \\sin\\alpha\\cos\\beta + \\cos\\alpha\\sin\\beta$ and $\\cos(\\alpha + \\beta) = \\cos\\alpha\\cos\\beta - \\sin\\alpha\\sin\\beta$, let you express the trig value of a combined angle. Setting $\\alpha = \\beta$ produces the double-angle identities like $\\sin(2\\theta) = 2\\sin\\theta\\cos\\theta$. A worked simplification: $\\frac{\\sin^2\\theta}{1 - \\cos\\theta}$ becomes $\\frac{1 - \\cos^2\\theta}{1 - \\cos\\theta} = \\frac{(1 - \\cos\\theta)(1 + \\cos\\theta)}{1 - \\cos\\theta} = 1 + \\cos\\theta$.

Polar Coordinates and Polar Graphs

Polar coordinates locate a point by a directed distance and an angle rather than horizontal and vertical offsets. A point is written $(r, \\theta)$, where $r$ is the distance from the origin, called the pole, and $\\theta$ is the angle from the positive $x$-axis, called the polar axis. Conversion to rectangular coordinates uses $x = r\\cos\\theta$ and $y = r\\sin\\theta$. Going the other way, $r = \\sqrt{x^2 + y^2}$ and $\\tan\\theta = \\frac{y}{x}$, with care taken to place $\\theta$ in the correct quadrant. Polar representations are not unique: the same point can be named with $\\theta + 2\\pi k$, and a negative $r$ points in the opposite direction, so $(r, \\theta)$ and $(-r, \\theta + \\pi)$ coincide. A polar function $r = f(\\theta)$ produces distinctive curves. The equation $r = a$ is a circle of radius $a$ centered at the pole, and $\\theta = c$ is a line through the pole. More elaborate graphs include $r = a + b\\cos\\theta$, which traces cardioids and limaicons, and $r = a\\cos(n\\theta)$, which traces rose curves. In AP Precalculus you analyze how $r$ changes as $\\theta$ increases: where $r$ is increasing the point moves farther from the pole, and the average rate of change $\\frac{\\Delta r}{\\Delta\\theta}$ describes that motion, mirroring the rate-of-change reasoning used throughout the course.