--- name: math-thumb-trigonometry description: "Apply Chapter 3 (Trigonometry: Control Angles, Directions, and Oscillations) of Mathematical Rules of Thumb to solve, check, or teach problems. Use it to choose angle representations that preserve quadrants and quantify a small-angle approximation." --- # Trigonometry: Control Angles, Directions, and Oscillations Use this chapter to help the reader make a checked mathematical decision. All 10 numbered rules are available in [the chapter source](references/chapter.md). [The workbook](references/notebook.md) contains a lab, exercises, solutions, and the full rule checklist. [The local rule index](references/rules.json) supplies discovery metadata. ## Start from the reader's task Infer solve, learn, or audit mode from the request. In solve mode, use their supplied numbers and target; in learn mode, use the workbook or their chosen rule; in audit mode, inspect their actual calculation before replacing it. Gather only missing information that changes the choice: Angle units; signed coordinates; oscillation frequency; triangle data type; tolerance near zero. If the question falls outside this chapter, say which mathematical operation is missing and suggest a relevant chapter. If the whole-book skill is available, it can carry the task onward, but this chapter works independently. ## Select and apply a rule - **Is the angle small?** Convert to radians. Use sine or tangent as $x$ only after checking the cubic correction; use $1-x^2/2$ when cosine's small departure from one matters. - **Are two signed components defining a direction or phase?** Preserve both with `atan2`. If they multiply same-frequency cosine and sine terms, convert the coefficient pair to amplitude and phase. - **Is the angle large or outside a familiar interval?** Reduce it modulo the function's period before using a reference angle or numerical routine. - **Is the triangle given by SAS or SSS?** Start with the law of cosines. If it is given by SSA, use the law of sines and test both inverse-sine branches. - **Does the formula subtract nearly equal trigonometric values?** Search for an exact identity such as $1-\cos x=2\sin^2(x/2)$ before evaluating. Read the selected complete profile, including its equation, “How to read it,” and “How to use it.” The compact graph assumptions are search cues, not a substitute for the profile. Preserve the numbered citation and role. **Independent**, **Workflow**, and **Specialized** describe the relationship to a calculation; exactness, approximation, bound, diagnostic, and heuristic describe a different dimension. Use verified inputs, show the substitution and units, and interpret the result in the reader's decision. Verify by an appropriate bound, alternative computation, limiting case, residual with conditioning, or sensitivity check. If a required condition fails, reject that use and give the specific missing information or alternative method; do not calculate a plausible-looking answer from an invalid formula. **Essential boundary:** Use radians for local models. Keep signed components for direction; check both inverse-sine branches for SSA triangle data. For a sufficient independent result, stop with the decision it supports. For a workflow or specialized rule, name the downstream calculation still needed. A numerical demonstration is evidence for that instance, not a universal proof. ## Teach and check understanding Use the [workbook](references/notebook.md) for guided practice. Start with 3.1.1, 3.2.1, 3.3.4 when the reader wants a starting exercise. Ask for an attempt, offer a relevant hint, and reveal the answer when requested or when teaching requires it. Do not force a quiz when the reader asked for a worked solution. Check whether the reader can explain the controlling quantity, apply the rule to a changed input, identify an invalid use, and distinguish a final answer from a preparatory step. Track only demonstrated work. Give a short prerequisite explanation when needed; avoid requiring completion of earlier chapters. ## Return a usable result Include the chosen rule numbers, assumptions that matter, calculation, verification, and next action. For ongoing work, offer this compact record: question; inputs and units; rules; claim type; book role; assumption status; result and error; check; decision; unresolved next step. Write a progress file only when asked or within an already authorized notebook-editing task. The source chapter is a fixed book snapshot. Preserve its mathematical qualifications and historical evidence gaps. Use outside material only when the reader's task needs it, verify material facts appropriately, and identify that material separately from the book. ## Illustrated exploration Open [the browser reader](assets/reader.html) or [the saved illustrated notebook](assets/notebook.ipynb). [Equation cards](references/equations.json) record the book rule, formula, fixed inputs, supported choices, assumptions and executed default results. - **C03-D01: Put an error budget on a small angle**: rule 3.1.1. - **C03-D02: Keep direction when a ratio loses the quadrant**: rule 3.2.1. - **C03-D03: Avoid subtracting nearly equal floating numbers**: rule 3.3.4. - **C03-D04: Count the triangles in the sine-law ambiguous case**: rule 3.3.2. Use a saved illustration only when its conditions fit. Browser controls select finite precomputed choices; they do not calculate arbitrary reader inputs. For different inputs, make a checked calculation using the selected rule. Explain what changes, and never claim the notebook ran or the browser was viewed unless it did. Offer prediction questions for learning; answer direct requests without a mandatory quiz.