# Chapter 3 notebook: Trigonometry: Control Angles, Directions, and Oscillations

**Goal:** Choose angle representations that preserve quadrants and quantify a small-angle approximation.

**Start with:** Rules 3.1.1, 3.2.1, 3.3.4. Use a calculator or the optional Python lab. Programming is optional for the workbook; the Jupyter version requires a Python 3 kernel. All lab numbers are constructed practice inputs, not observed data.

**Source:** [Finished Chapter 3](../skills/math-thumb-trigonometry/references/chapter.md) from *Mathematical Rules of Thumb*. Numbered rules, classifications, and the decision path come from the book. The lab, exercises, answer key, and worksheet prompts are companion additions.

## 1. Frame your decision

Write a question in which an answer would change something you do. Gather: Angle units; signed coordinates; oscillation frequency; triangle data type; tolerance near zero.

- My question and intended decision:
- Known inputs and units:
- Required accuracy or threshold:
- What I expect before calculating:
- What I still need to find out:

Use the lab as a worked starting point if you do not yet have your own problem. For notation or prerequisites, ask the chapter skill to explain only the concept blocking the next step.

## 2. Choose a route

- **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.

**Chapter-specific stop check:** Use radians for local models. Keep signed components for direction; check both inverse-sine branches for SSA triangle data.

## 3. Work the lab

For an angle of 5 degrees, estimate sine by the angle in radians. Convert first: x=5 × pi/180, about 0.0872665. Sine is about 0.0871557; the absolute error is about 0.0001107. Taylor's bound |sin(x)-x| <= |x|^3/6 certifies this case. If the required absolute error is 0.0001, the approximation is insufficient.

Predict the sign and scale before running the code. Then change one input and explain why the result moves. The code checks the constructed example; it does not prove the rule for every possible input. Code assertions may describe the example's chosen regime, so inspect them before changing that regime.

```python
import math
degrees, tolerance = 5.0, 0.0001
x = math.radians(degrees)
exact = math.sin(x)
error_bound = abs(x)**3/6
error = abs(exact-x)
print(f"Radians: {x:.8f}; sine: {exact:.8f}; error: {error:.8f}")
print(f"Bound: {error_bound:.8f}; tolerance met: {error <= tolerance}")
assert error <= error_bound + 1e-15
```

**My prediction, observed result, and explanation:**

_Record your work here._

## 4. Practise without the answers

### Exercise 1

Find the direction of the vector (-1,1), measured counterclockwise from positive x.

**My approach, assumptions, calculation, and check:**

_Write your attempt here._

### Exercise 2

For x=0.001 radians, estimate 1-cos(x) and name a stable evaluation form.

**My approach, assumptions, calculation, and check:**

_Write your attempt here._

### Exercise 3

A student substitutes 10 directly into sin(x)≈x for a 10-degree angle. Identify and repair the failure.

**My approach, assumptions, calculation, and check:**

_Write your attempt here._

**Coaching prompt:** “Use the Chapter 3 skill to help me with Exercise 2. Ask for my attempt, give one useful hint if I need it, and help me check the assumptions before showing the answer.”

## 5. Answer key and reasoning

Read this after attempting the exercises, or use it immediately if you prefer a complete walkthrough. An answer is complete only when its assumptions and stopping point are clear.

### Answer 1

atan2(1,-1)=3 × pi/4=135 degrees. atan(1/-1) returns -45 degrees and loses the quadrant.

### Answer 2

The leading estimate is x^2/2=5e-7. Evaluate 2 × sin(x/2)^2 to avoid subtracting nearly equal numbers.

### Answer 3

The input must be radians: 10 × pi/180≈0.174533. The estimate is 0.174533, compared with sin(10 degrees)≈0.173648. Whether that is sufficient depends on the tolerance.

## 6. Build the complete chapter toolkit

The new lab samples the chapter; the following checklist covers all 10 rules. Study one thematic group at a time. A large group can take several sessions.

- **Use Small Angles as Local Linear Models:** work with rules 3.1.1, 3.1.2, 3.1.3.
- **Preserve Direction and Periodic Structure:** work with rules 3.2.1, 3.2.2, 3.2.3.
- **Solve Triangles Without Losing Cases or Precision:** work with rules 3.3.1, 3.3.2, 3.3.3, 3.3.4.

For each selected rule, read its equation, explanation, and worked use in the source. Reproduce that example; change one input; then change one assumption so the rule is no longer justified. Record the result and what check catches the failure. Historical examples remain labeled and qualified as in the source.

Read each complete numbered profile in the [chapter reference](../skills/math-thumb-trigonometry/references/chapter.md) before applying it. The cues below abbreviate the graph metadata; they are not complete conditions. Change the status only after doing the practice described below.

| Rule | Book role | First assumptions to inspect | Practice status |
|---|---|---|---|
| 3.1.1: Replace sine by the angle at small scale | Independent | small angle;  radian measure | new |
| 3.1.2: Keep the quadratic correction for cosine near zero | Independent | small angle;  radian measure | new |
| 3.1.3: Use tangent approximately equal to angle only away from poles | Independent | small angle;  radian measure | new |
| 3.2.1: Use atan2, not a plain arctangent, for direction | Independent | nonzero vector;  documented argument order | new |
| 3.2.2: Combine a sine-cosine pair into one shifted sinusoid | Independent | same frequency and argument;  real coefficients | new |
| 3.2.3: Reduce large angles before evaluating trigonometric functions | Workflow | known period;  angle reduction done with sufficient precision | new |
| 3.3.1: View the law of cosines as corrected Pythagoras | Independent | euclidean triangle;  included angle matches two known sides | new |
| 3.3.2: Check both branches in the sine-law ambiguous case | Workflow | side side angle data;  known angle opposite known side | new |
| 3.3.3: Use radians whenever angles enter calculus or approximation | Workflow | angular quantity;  local change or calculus formula | new |
| 3.3.4: Use a half-angle identity for one minus cosine | Workflow | consistent angle units | new |

For a completed row, record: **rule number / my new input / mathematical claim type / verified assumptions / calculation / check / valid use / rejected use / next step**. “Practised” means you worked an example. “Demonstrated” means you can explain a valid use, transfer it, and reject a misuse without the answer key.

## 7. Apply it to your own problem

Return to your opening question. Choose the smallest rule set that can settle it. Use the book's independent/workflow/specialized classification separately from the claim type (exact, approximate, bound, diagnostic, or heuristic).

- Selected rule number(s) and reason:
- Assumptions that hold, fail, or remain uncertain:
- Substitution with units:
- Result and error, uncertainty, or bound:
- Independent check or limiting case:
- Decision this supports:
- Stop here, do a named next calculation, or gather missing information:

## 8. Transfer and continue

Useful nearby chapters: Chapter 4: calculus; Chapter 24: signal processing; Chapter 25: control theory. Bring the question, units, assumptions, result type, and uncertainty to the next chapter. Choose a bridge only when it supplies an operation you actually need.

**Completion check:** Explain this chapter's lab in your own words; solve one changed-input exercise; reject one invalid use; and produce a decision record for your own problem. If one check fails, revisit that part of the chapter rather than marking every rule complete.