Precalculus Reference Sheet

\(\sin(x \pm y) = \sin(x)\cos(y) \pm \cos(x)\sin(y)\)

\(\cos(x \pm y) = \cos(x)\cos(y) \mp \sin(x)\sin(y)\)

\(\tan(x \pm y) = \frac{\tan(x) \pm \tan(y)}{1 \mp \tan(x)\tan(y)}\)

\(\sin\left(x \pm \frac{\pi}{2}\right) = \pm \cos(x)\)

\(\cos\left(x \pm \frac{\pi}{2}\right) = \mp \sin(x)\)

\(\sin(2x) = 2\sin(x)\cos(x)\)

\(\cos(2x) = \cos^2(x) - \sin^2(x) = 1 - 2\sin^2(x) = 2\cos^2(x) - 1\)

\(\tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)}\)

\(\sin^2(x) = \frac{1 - \cos(2x)}{2}\)

\(\cos^2(x) = \frac{1 + \cos(2x)}{2}\)

\(\tan^2(x) = \frac{1 - \cos(2x)}{1 + \cos(2x)}\)

\(\sin\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 - \cos(x)}{2}}\)

\(\cos\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 + \cos(x)}{2}}\)

\(\tan\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 - \cos(x)}{1 + \cos(x)}} = \frac{\sin(x)}{1 + \cos(x)} = \frac{1 - \cos(x)}{\sin(x)}\)

\(\sin^{-1}(x) = \arcsin(x)\), where \(x = \sin(y)\)

\(\cos^{-1}(x) = \arccos(x)\), where \(x = \cos(y)\)

\(\tan^{-1}(x) = \arctan(x)\), where \(x = \tan(y)\)

\(\sin(\sin^{-1}(x)) = x\)

\(\cos(\cos^{-1}(x)) = x\)

\(\tan(\tan^{-1}(x)) = x\)

\(\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} = 2R\)

Where \(R\) is the radius of the circumcircle of the triangle

Area Formulas

\(\text{Area} = \frac{1}{2}bc\sin(A) = \frac{1}{2}ab\sin(C) = \frac{1}{2}ac\sin(B)\)

Use Law of Sines for:

AAS, ASA, SSA (ambiguous case)

\(a^2 = b^2 + c^2 - 2bc\cos(A)\)

\(\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}\)

\(A = \cos^{-1}\left(\frac{b^2 + c^2 - a^2}{2bc}\right)\)

Use Law of Cosines for:

SAS, SSS

Semi-perimeter: \(s = \frac{a + b + c}{2}\)

\(\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}\)

Inradius: \(r = \frac{\sqrt{(s-a)(s-b)(s-c)}}{s}\)

A complex number: \(Z = a + bi\)

Where \(a\) is the real part and \(b\) is the imaginary part

\(|Z| = |a + bi| = \sqrt{a^2 + b^2}\)

Example: \(|3 + 4i| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5\)

\(a = r\cos(\theta)\)

\(b = r\sin(\theta)\)

\(r = |Z| = \sqrt{a^2 + b^2}\) (distance to origin)

\(a + bi = r(\cos(\theta) + i\sin(\theta)) = r\text{cis}(\theta)\)

Example: \(4 + 4i = 4\sqrt{2}\text{cis}\left(\frac{\pi}{4}\right)\)

\(r_1\text{cis}(\theta_1) \cdot r_2\text{cis}(\theta_2) = r_1r_2\text{cis}(\theta_1 + \theta_2)\)

\(\frac{r_1\text{cis}(\theta_1)}{r_2\text{cis}(\theta_2)} = \frac{r_1}{r_2}\text{cis}(\theta_1 - \theta_2)\)

\((r\text{cis}(\theta))^n = r^n\text{cis}(n\theta)\)

Example: \((1+i)^9 = \sqrt{2}\text{cis}(45°)^9 = 2^{9/2}\text{cis}(405°) = 16\sqrt{2}\text{cis}(45°) = 16 + 16i\)

\((r\text{cis}(\theta))^{1/n} = r^{1/n}\text{cis}\left(\frac{\theta + k \cdot 360°}{n}\right)\)

Where \(k = 0, 1, 2, \ldots, n-1\)

If an event can be performed in \(n_1\) ways, a second in \(n_2\) ways, etc., then the total number of ways is:

\(n_1 \cdot n_2 \cdot n_3 \cdot \ldots \cdot n_k\)

Example: A 5-digit number has \(10 \cdot 10 \cdot 10 \cdot 10 \cdot 10 = 100,000\) combinations

Permutation of all n objects:

\(_nP_n = n(n-1)(n-2)\ldots(3)(2)(1) = n!\)

Permutations of n objects taken k at a time:

\(_nP_k = \frac{n!}{(n-k)!}\)

Order matters (abc ≠ bac)

\(_nC_k = \binom{n}{k} = \frac{n!}{k!(n-k)!}\)

Order does not matter (abc = bac = cab)

Used to expand \((a + b)^n\):

\((a + b)^0 = 1\)

\((a + b)^1 = a + b\)

\((a + b)^2 = a^2 + 2ab + b^2\)

\((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\)

\((a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4\)

Key Identity:

\(\binom{n}{k} = \binom{n}{n-k}\)

If \(f(x) = mx + b\), then \(\lim_{x \to a} f(x) = ma + b\)

If \(f(x)\) has a limit as \(x\) approaches \(a\), that limit is unique.

If \(\lim_{x \to a} f(x) = L_1\) and \(\lim_{x \to a} g(x) = L_2\), then \(\lim_{x \to a} [f(x) + g(x)] = L_1 + L_2\)

If \(\lim_{x \to a} f(x) = L_1\) and \(\lim_{x \to a} g(x) = L_2\), then \(\lim_{x \to a} [f(x) \cdot g(x)] = L_1 \cdot L_2\)

Corollary 1:

\(\lim_{x \to a} [k \cdot f(x)] = k \cdot L_1\) (where \(k\) is a constant)

Corollary 2:

\(\lim_{x \to a} x^n = a^n\)

Corollary 3:

If \(f(x) = c_0x^n + c_1x^{n-1} + c_2x^{n-2} + \ldots + c_n\), then \(\lim_{x \to a} f(x) = c_0a^n + c_1a^{n-1} + c_2a^{n-2} + \ldots + c_n\)

If \(\lim_{x \to a} g(x) = L \neq 0\), then \(\lim_{x \to a} \frac{1}{g(x)} = \frac{1}{L}\)

Corollary 1:

If \(\lim_{x \to a} f(x) = L_1\) and \(\lim_{x \to a} g(x) = L_2 \neq 0\), then \(\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L_1}{L_2}\)

\(\frac{df(x)}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\)

Also written as \(f'(x)\) (Lagrange notation)

Example: \(f(x) = x^2\)

\(\frac{df}{dx} = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h} = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h} = \lim_{h \to 0} \frac{2xh + h^2}{h} = \lim_{h \to 0} (2x + h) = 2x\)

\(\frac{d}{dx}(a^x) = \ln(a) \cdot a^x\)

Proof: Take logarithm of \(y = a^x\) and use implicit differentiation

\(\frac{d}{dx}[k \cdot f(x)] = k \cdot \frac{d}{dx}f(x)\)

Example: \(\frac{d}{dx}(5\sin x) = 5\cos x\)

\(\frac{d}{dx}[f(x) \cdot g(x)] = f(x) \cdot \frac{d}{dx}g(x) + \frac{d}{dx}f(x) \cdot g(x)\)

Or: \((fg)' = f'g + g'f\)

\(\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{\frac{d}{dx}f(x) \cdot g(x) - f(x) \cdot \frac{d}{dx}g(x)}{g^2(x)}\)

Or: \(\left(\frac{f}{g}\right)' = \frac{f'g - g'f}{g^2}\)

\(\frac{d}{dx}\left[\frac{1}{f(x)}\right] = -\frac{\frac{d}{dx}f(x)}{f^2(x)}\)

Or: \(\left(\frac{1}{f}\right)' = -\frac{f'}{f^2}\) (derived from quotient rule)

\([f(g(x))]' = f'(g(x)) \cdot g'(x)\)

Example: \(\frac{d}{dx}[\sin(x^3)] = \cos(x^3) \cdot 3x^2 = 3x^2\cos(x^3)\)

1. \(f(x) = 6x^2 + 7x\)

2. \(g(t) = (4t^2 - 3t + 2)^{-2}\)

3. \(y = \frac{3}{1-8z}\)

4. \(R(w) = \csc(7w)\)

5. \(G(x) = 2\sin(3x) + \tan(x)\)

6. \(h(u) = \tan(4 + 10u)\)

7. \(f(t) = 5 + e^{4t} + t^7\)

8. \(g(x) = e^{1-\cos x}\)

9. \(H(z) = \frac{2}{1-6z}\)

10. \(u(t) = \tan^{-1}(3t-1)\)