Trigonometry Study Guide

📖 Core Concepts Trigonometry – studies how angles relate to side lengths in triangles; extends to periodic functions on the unit circle. Right‑triangle ratios – sine = opp/hyp, cosine = adj/hyp, tangent = opp/adj (SOH‑CAH‑TOA). Unit circle – radius 1 centered at the origin; point $(\cos\theta,\sin\theta)$ gives cosine and sine for any real angle. Reciprocal functions – cosecant = $1/\sin$, secant = $1/\cos$, cotangent = $1/\tan$; “co‑” means the complement (e.g., $\csc\theta=\sin(90^\circ\!-\!\theta)$). Periodicity – $\sin$ and $\cos$ repeat every $2\pi$, $\tan$ every $\pi$. Inverse trig – $\arcsin$, $\arccos$, $\arctan$ are the one‑to‑one inverses after restricting domains. 📌 Must Remember SOH‑CAH‑TOA: $\sin\theta=\frac{\text{opp}}{\text{hyp}}$, $\cos\theta=\frac{\text{adj}}{\text{hyp}}$, $\tan\theta=\frac{\text{opp}}{\text{adj}}$. Pythagorean identity: $\sin^{2}\theta+\cos^{2}\theta=1$. Derived identities: $\tan^{2}\theta+1=\sec^{2}\theta$, $1+\cot^{2}\theta=\csc^{2}\theta$. Key unit‑circle values | Angle | $\sin$ | $\cos$ | $\tan$ | |------|-------|-------|-------| | $0^\circ$ | $0$ | $1$ | $0$ | | $30^\circ$ | $\tfrac12$ | $\tfrac{\sqrt3}{2}$ | $\tfrac1{\sqrt3}$ | | $45^\circ$ | $\tfrac{\sqrt2}{2}$ | $\tfrac{\sqrt2}{2}$ | $1$ | | $60^\circ$ | $\tfrac{\sqrt3}{2}$ | $\tfrac12$ | $\sqrt3$ | Law of Sines: $\displaystyle \frac{a}{\sin A}= \frac{b}{\sin B}= \frac{c}{\sin C}=2R$. Law of Cosines: $\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos C$. Area formula: $\displaystyle \text{Area}= \tfrac12ab\sin C$. Angle‑sum: $\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$, $\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$. Half‑angle: $\displaystyle \sin\frac\theta2=\pm\sqrt{\frac{1-\cos\theta}{2}}$, $\displaystyle \cos\frac\theta2=\pm\sqrt{\frac{1+\cos\theta}{2}}$. 🔄 Key Processes Find a missing side or angle in a right triangle Identify the known angle. Choose the appropriate ratio (SOH‑CAH‑TOA). Rearrange to solve for the unknown. Solve any triangle (non‑right) List known sides/angles. Decide whether to use Law of Sines (two angles known or two sides + non‑included angle) or Law of Cosines (SSS or SAS). Apply the chosen law, solve for the missing quantity, then use the Law of Sines or angle‑sum to finish. Convert between degrees and radians (if needed): multiply by $\pi/180$ for degrees → radians, divide by $\pi/180$ for radians → degrees. Evaluate trig functions for special angles Memorize the 0°, 30°, 45°, 60°, 90° table. Use symmetry: $\sin(180^\circ-\theta)=\sin\theta$, $\cos(180^\circ-\theta)=-\cos\theta$, etc. Use identities to simplify expressions Replace $\sin^2+\cos^2$ with 1. Convert products to sums (product‑to‑sum) when helpful. 🔍 Key Comparisons Sine vs. Cosine $\sin\theta = \cos(90^\circ\!-\!\theta)$ Graph: sine starts at 0, cosine starts at 1 (phase shift of $\pi/2$). Tangent vs. Cotangent $\tan\theta = \frac{1}{\cot\theta}$, $\cot\theta = \tan(90^\circ\!-\!\theta)$. Asymptotes: $\tan$ at $\frac{\pi}{2}+k\pi$, $\cot$ at $k\pi$. Law of Sines vs. Law of Cosines Law of Sines → best when you have AAS or SSA (but watch for the ambiguous case). Law of Cosines → best for SSS or SAS; also resolves the ambiguous case. Inverse vs. Reciprocal $\arcsin x$ gives an angle; $\csc\theta = 1/\sin\theta$ gives a ratio. ⚠️ Common Misunderstandings Mixing up opposite/adjacent for a given acute angle – always label the triangle first. Using $\sin(90^\circ-\theta)=\sin\theta$ – false; it equals $\cos\theta$. Assuming Law of Sines always works for SSA – can produce two possible triangles; check feasibility with the altitude test. Treating $\tan\theta$ as $\sin\theta/\cos\theta$ without considering cos = 0 – leads to division‑by‑zero errors at odd multiples of $90^\circ$. Neglecting sign in half‑angle formulas – choose “+” for quadrants I & II (sine) or I & IV (cosine), “–” otherwise. 🧠 Mental Models / Intuition Unit circle as a GPS: the x‑coordinate = cosine (horizontal), y‑coordinate = sine (vertical). Rotate the radius to visualize sign changes across quadrants. SOH‑CAH‑TOA as “height‑base‑hypotenuse” – think of the triangle sitting on a flat base; opposite = “height,” adjacent = “base.” Law of Cosines = Pythagoras + correction – when the included angle isn’t $90^\circ$, subtract $2ab\cos C$ to account for the “tilt.” Periodicity = clock face – every $2\pi$ (or 360°) the sine/cosine hands point to the same spot; tangent repeats every half‑turn ($\pi$). 🚩 Exceptions & Edge Cases Undefined values: $\sec\theta$, $\csc\theta$, $\cot\theta$ are undefined where $\cos\theta$, $\sin\theta$, $\tan\theta$ are zero, respectively. Inverse domain limits: $\arcsin x$ and $\arccos x$ only accept $x\in[-1,1]$. Ambiguous SSA case: if $a < b$ and $a > b\sin A$, two triangles exist; if $a = b\sin A$, one right triangle; if $a \le b\sin A$, no triangle. Half‑angle sign: depends on the quadrant of $\frac{\theta}{2}$. 📍 When to Use Which Right‑triangle problems → SOH‑CAH‑TOA directly. Non‑right triangle, two angles known → Law of Sines (quick). Non‑right triangle, two sides and included angle → Law of Cosines (or directly compute side). Non‑right triangle, three sides known → Law of Cosines to find any angle, then Law of Sines if needed. Area needed, two sides + included angle → $\frac12ab\sin C$. Simplifying algebraic trig expressions → use Pythagorean, sum‑to‑product, or double‑angle identities first. 👀 Patterns to Recognize “1 – cos θ” or “1 + cos θ” under a square root → half‑angle form. Products of sines or cosines → look for product‑to‑sum conversion. Expression of the form $a\sin\theta + b\cos\theta$ → can be rewritten as $R\sin(\theta+\phi)$. Repeated $ \sin^2 + \cos^2$ → replace with 1. Angles differing by $90^\circ$ → swap sine and cosine (or use reciprocal “co‑” functions). 🗂️ Exam Traps Distractor: $\tan 30^\circ = \frac{1}{\sqrt{3}}$ vs. $\frac{\sqrt{3}}{3}$ – both are equivalent; answer may use either form. Choosing the wrong quadrant for inverse functions – e.g., $\arcsin(0.5)$ is $30^\circ$, not $150^\circ$ (out of range). Assuming $\sin(2\theta)=2\sin\theta$ – correct is $2\sin\theta\cos\theta$ (double‑angle). Mixing up law of sines vs. law of cosines in SSA – if you apply Cosines blindly you’ll get an incorrect side. Half‑angle sign error – many answer keys forget the “±”; check the quadrant of $\theta/2$. Misreading “adjacent” as the side opposite the given angle – always draw a quick diagram to label. --- Study tip: After reviewing each section, close the book and write down the core formulas from memory. Then solve one problem of each type (right‑triangle, SSA, SAS, area) to cement the process. Good luck!