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Pre-Calculus Reference Sheet

One page to keep beside you all year — the unit circle, the identities, the function families, the conics, and the series that lead into calculus. Built to print cleanly on a single light background.

Unit Circle Identities Function Families Conics Series Polar & Parametric

Everything below is exact, not approximate. Use it to check your work, not to skip the thinking — knowing where each value or formula comes from is the point of the year.

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The Unit Circle — Exact Values

On a circle of radius \(1\), the point at angle \(\theta\) is \((\cos\theta,\ \sin\theta)\), and \(\tan\theta=\dfrac{\sin\theta}{\cos\theta}\).

First-quadrant diagram

Radian map for the first quadrant: \(0,\ \dfrac{\pi}{6},\ \dfrac{\pi}{4},\ \dfrac{\pi}{3},\ \dfrac{\pi}{2}\). The other three quadrants repeat these reference angles with sign changes — cosine is the \(x\)-coordinate, sine is the \(y\)-coordinate.

First-quadrant exact values
\(\theta\)	\(\sin\theta\)	\(\cos\theta\)	\(\tan\theta\)
\(0\)	\(0\)	\(1\)	\(0\)
\(\dfrac{\pi}{6}\)	\(\dfrac{1}{2}\)	\(\dfrac{\sqrt3}{2}\)	\(\dfrac{\sqrt3}{3}\)
\(\dfrac{\pi}{4}\)	\(\dfrac{\sqrt2}{2}\)	\(\dfrac{\sqrt2}{2}\)	\(1\)
\(\dfrac{\pi}{3}\)	\(\dfrac{\sqrt3}{2}\)	\(\dfrac{1}{2}\)	\(\sqrt3\)
\(\dfrac{\pi}{2}\)	\(1\)	\(0\)	und.

Quadrant signs (ASTC): all positive in QI; only \(\sin\) (and \(\csc\)) in QII; only \(\tan\) (and \(\cot\)) in QIII; only \(\cos\) (and \(\sec\)) in QIV. The quadrantal points are \((1,0),(0,1),(-1,0),(0,-1)\) at \(0,\dfrac{\pi}{2},\pi,\dfrac{3\pi}{2}\).

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Core Trigonometric Identities

True for every angle where both sides are defined — the algebra of the circle.

Reciprocal

\(\csc\theta = \dfrac{1}{\sin\theta}, \quad \sec\theta = \dfrac{1}{\cos\theta}, \quad \cot\theta = \dfrac{1}{\tan\theta}\)

Quotient

\(\tan\theta = \dfrac{\sin\theta}{\cos\theta}, \qquad \cot\theta = \dfrac{\cos\theta}{\sin\theta}\)

Pythagorean

\(\sin^2\theta + \cos^2\theta = 1\)
\(1 + \tan^2\theta = \sec^2\theta\)
\(1 + \cot^2\theta = \csc^2\theta\)

Even – Odd

\(\cos(-\theta) = \cos\theta\) (even), \(\sec(-\theta)=\sec\theta\)
\(\sin(-\theta) = -\sin\theta\) (odd), \(\tan(-\theta)=-\tan\theta\)

Cofunction

\(\sin\!\left(\tfrac{\pi}{2}-\theta\right) = \cos\theta, \quad \cos\!\left(\tfrac{\pi}{2}-\theta\right) = \sin\theta\)
\(\tan\!\left(\tfrac{\pi}{2}-\theta\right) = \cot\theta\)

Sum & Difference

\(\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B\)
\(\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B\)
\(\tan(A \pm B) = \dfrac{\tan A \pm \tan B}{1 \mp \tan A\tan B}\)

Double-Angle

\(\sin 2\theta = 2\sin\theta\cos\theta\)
\(\cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta\)
\(\tan 2\theta = \dfrac{2\tan\theta}{1 - \tan^2\theta}\)

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Function-Family Summary

Each parent function, its rule, and the signature you read at a glance. Transform any of them with \(a\,f\!\left(b(x-h)\right)+k\).

Parent functions & their key attributes
Family	Parent	Domain	Range	Signature feature
Linear	\(f(x)=x\)	\(\mathbb{R}\)	\(\mathbb{R}\)	constant slope
Quadratic	\(f(x)=x^2\)	\(\mathbb{R}\)	\([0,\infty)\)	vertex, axis of symmetry
Cubic	\(f(x)=x^3\)	\(\mathbb{R}\)	\(\mathbb{R}\)	inflection at origin, odd
Absolute value	\(f(x)=\|x\|\)	\(\mathbb{R}\)	\([0,\infty)\)	V-shape, even
Square root	\(f(x)=\sqrt{x}\)	\([0,\infty)\)	\([0,\infty)\)	endpoint, half-curve
Rational	\(f(x)=\dfrac{1}{x}\)	\(x\neq 0\)	\(y\neq 0\)	asymptotes \(x=0,\,y=0\)
Exponential	\(f(x)=b^{x}\), \(b>0\)	\(\mathbb{R}\)	\((0,\infty)\)	asymptote \(y=0\); grows/decays
Logarithmic	\(f(x)=\log_b x\)	\((0,\infty)\)	\(\mathbb{R}\)	asymptote \(x=0\); inverse of \(b^x\)
Sine	\(f(x)=\sin x\)	\(\mathbb{R}\)	\([-1,1]\)	period \(2\pi\), odd
Cosine	\(f(x)=\cos x\)	\(\mathbb{R}\)	\([-1,1]\)	period \(2\pi\), even

Transformations. In \(y = a\,f\!\left(b(x-h)\right) + k\): \(a\) is a vertical stretch (and reflection if \(a<0\)); \(b\) is a horizontal stretch by \(\tfrac{1}{b}\) (for sinusoids the period is \(\tfrac{2\pi}{|b|}\)); \(h\) shifts right; \(k\) shifts up. Logarithm laws: \(\log_b(MN)=\log_b M+\log_b N\), \(\log_b\!\tfrac{M}{N}=\log_b M-\log_b N\), \(\log_b M^{p}=p\log_b M\), and change of base \(\log_b M=\dfrac{\ln M}{\ln b}\).

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Conic Sections — Standard Forms

Center or vertex at \((h,k)\). Each conic is the set of points obeying a distance rule.

Parabola

Vertex \((h,k)\), \(p=\) vertex-to-focus distance

Opens up / down:

\[ (x-h)^2 = 4p\,(y-k) \]

focus \((h,\ k+p)\), directrix \(y=k-p\).

Opens right / left:

\[ (y-k)^2 = 4p\,(x-h) \]

focus \((h+p,\ k)\), directrix \(x=h-p\).

Ellipse

Center \((h,k)\), \(a>b>0\), \(c^2=a^2-b^2\)

Horizontal major axis:

\[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \]

Vertical major axis:

\[ \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 \]

Vertices are \(a\) from center along the major axis; foci are \(c\) from center.

Hyperbola

Center \((h,k)\), \(c^2=a^2+b^2\)

Opens left / right:

\[ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \]

asymptotes \(y - k = \pm\dfrac{b}{a}(x-h)\).

Opens up / down:

\[ \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 \]

asymptotes \(y - k = \pm\dfrac{a}{b}(x-h)\).

Vertices are \(a\) from center along the transverse axis; foci are \(c\) from center, with \(c>a\). A circle is the special ellipse \((x-h)^2+(y-k)^2=r^2\).

∞

Sequences & Series

The patterns that foreshadow the infinite — and the first thing calculus formalizes with limits.

Arithmetic (common difference \(d\))

\(n\)th term: \(a_n = a_1 + (n-1)d\)
Partial sum: \(\displaystyle S_n = \frac{n}{2}\,(a_1 + a_n) = \frac{n}{2}\bigl(2a_1+(n-1)d\bigr)\)

Geometric (common ratio \(r\))

\(n\)th term: \(a_n = a_1\,r^{\,n-1}\)
Partial sum: \(\displaystyle S_n = a_1\,\frac{1 - r^{n}}{1 - r}, \quad r \neq 1\)

Infinite geometric

Converges only when \(|r| < 1\): \(\displaystyle S_\infty = \sum_{k=1}^{\infty} a_1\,r^{\,k-1} = \frac{a_1}{1 - r}\)

Sigma (summation) notation

\(\displaystyle \sum_{k=1}^{n} a_k = a_1 + a_2 + \cdots + a_n\)
\(\displaystyle \sum_{k=1}^{n} c = c\,n, \qquad \sum_{k=1}^{n} k = \frac{n(n+1)}{2}\)
\(\displaystyle \sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}\)

Binomial Theorem

\(\displaystyle (a+b)^{n} = \sum_{k=0}^{n} \binom{n}{k} a^{\,n-k} b^{\,k}\)
where \(\displaystyle \binom{n}{k} = \frac{n!}{k!\,(n-k)!}\) — the entries of Pascal's triangle.

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Polar & Parametric Conversions

Two other ways to name a point or trace a curve — by direction-and-distance, or by a parameter \(t\).

Polar ↔ Rectangular

A point as \((r,\theta)\) instead of \((x,y)\)

Polar to rectangular:

\[ x = r\cos\theta, \qquad y = r\sin\theta \]

Rectangular to polar:

\[ r^2 = x^2 + y^2, \qquad \tan\theta = \frac{y}{x} \]

Choose \(\theta\) by the quadrant of \((x,y)\); \(r\) may be taken positive.

Parametric Curves

Both coordinates driven by a parameter \(t\)

A curve given by

\[ x = x(t), \qquad y = y(t) \]

Eliminate the parameter by solving one equation for \(t\) and substituting, or by using an identity. For example, \(x=\cos t,\ y=\sin t\) gives

\[ x^2 + y^2 = \cos^2 t + \sin^2 t = 1, \]

the unit circle — traced counterclockwise as \(t\) increases.

Keep this sheet close. It pairs with each module's Visual Lab on the Pre-Calculus course page — use the labs to see a curve move, and use this sheet to write the exact values and forms behind it.