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Lesson 20 — Trim 2 Consolidation: trigonometry + sequences workshop

Integrating workshop for lessons 11-19. Problems combining trigonometry, sequences and intuitive limit.

Used in: 1.º ano do EM (15 anos) · Equiv. Math II japonês — revisão de unidade · Equiv. Klasse 10 alemã — Abschlusstest · Equiv. O-Level Singapore — End-of-topic consolidation

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Rigorous notation, full derivation, hypotheses

Trim 2 Roadmap

Workshop with 25 problems that require combining:

Lesson 11: trigonometric ratios in a right triangle
Lesson 12: trigonometric circle, radians
Lesson 13: trigonometric functions and modeling
Lesson 14: trigonometric equations and inequalities
Lesson 15: laws of sines and cosines
Lessons 16-18: sequences, AP, GP
Lesson 19: intuitive limit

Problem styles

Combined application (10): trigonometry + function
Modeling (8): translating real-world statements
Challenge (5): hard ENEM or Olympiad level
Proof (2): consolidating demonstrations

Set aside 4h with no consultation to solve. Check the answer key (25% have a developed solution). If you score < 50%, reread the corresponding lessons; 70-90% ready for Trim 3; > 90%, additional reading from reference books.

Exercise list

25 exercises · 6 with worked solution (25%)

Application 10Understanding 1Modeling 7Challenge 5Proof 2

Ex. 20.1ApplicationAnswer key
Solve $\sin(2x) = 1/2$ on $[0, 2\pi)$ .
Solve online
Ex. 20.2Application
Triangle with $a = 5$ , $A = 30°$ , $B = 60°$ . Compute $b$ using the law of sines.
Solve online
Ex. 20.3Application
Triangle with sides $7, 8, 9$ . Largest angle?
Solve online
Ex. 20.4Application
Sketch $y = 2\sin(\pi x/3)$ — period and amplitude.
Solve online
Ex. 20.5Application
Compute $\sin(\pi/3)$ , $\cos(7\pi/4)$ , $\tan(5\pi/6)$ .
Solve online
Ex. 20.6ModelingAnswer key
Tide in Salvador: $h(t) = 1{,}5 + 1\sin(\pi t/6)$ . When is $h = 1{,}5$ ? When is it maximum?
Solve online
Ex. 20.7Modeling
Grid voltage: $V(t) = 311 \sin(120\pi t)$ V. RMS voltage: $V_{ef} = V_0/\sqrt 2$ . Compute approximate $V_{ef}$ (~220 V).
Solve online
Ex. 20.8Modeling
Topography: you are 100 m from a tower, elevation angle $30°$ . Height?
Solve online
Ex. 20.9Modeling
Solve $\cos x + \sin x = 1$ on $[0, 2\pi)$ . (Use $\cos x + \sin x = \sqrt 2 \sin(x + \pi/4)$ .)
Solve online
Ex. 20.10Challenge
Prove that in an equilateral triangle of side $\ell$ , the area $= \ell^2 \sqrt 3 /4$ via the law of cosines.
Ex. 20.11Application
AP with $a_1 = 3$ , $r = 5$ . Compute $a_{20}$ and $S_{20}$ .
Solve online
Ex. 20.12ApplicationAnswer key
GP with $a_1 = 4$ , $q = 3$ . Compute $a_8$ and $S_8$ .
Solve online
Ex. 20.13Application
Compute $\sum_{n=0}^\infty (1/3)^n$ .
Solve online
Ex. 20.14ApplicationAnswer key
Determine whether the sequence $a_n = (n+1)/n$ converges. To what?
Solve online
Ex. 20.15Application
Insert 4 terms forming an AP between 5 and 25.
Solve online
Ex. 20.16ModelingAnswer key
You save R$ 100 in the first month and grow 5% per month. Balance after 12 months?
Solve online
Ex. 20.17Modeling
A ball drops from 10 m and after each bounce rises 70% of the height. Total distance traveled?
Solve online
Ex. 20.18Modeling
Inflation: 0.5% per month compounded. How much after 12 months?
Solve online
Ex. 20.19Understanding
Show that $0{,}999\ldots = 1$ via infinite GP.
Solve online
Ex. 20.20Challenge
Sequence $a_1 = 2$ , $a_{n+1} = (a_n + 5)/2$ . To which value does it converge?
Solve online
Ex. 20.21Challenge
In how much time does R$ 1,000 double at 6% p.a. with continuous compounding? (Use $\ln$ .)
Solve online
Ex. 20.22Challenge
Determine the general term and $S_n$ of the sequence $1, 4, 9, 16, 25, \ldots$ (it is neither AP nor GP).
Solve online
Ex. 20.23ChallengeAnswer key
Solve: $\sin x + \sin 2x + \sin 3x = 0$ on $[0, 2\pi)$ .
Solve online
Ex. 20.24Proof
Prove that in a triangle $ABC$ , $\sin A / a = \sin B / b$ using the law of cosines.
Ex. 20.25Proof
Prove that $\lim_{n \to \infty} (1 + 1/n)^n$ exists using monotonicity + boundedness. (Sketch; rigor comes in Analysis.)

Sources for this lesson

Consolidation lesson gathers sources from lessons 11-19. Main ones:

Algebra and Trigonometry — Jay Abramson et al. (OpenStax) · 2022, 2nd ed · EN · CC-BY · §8-11: trigonometry and sequences.
Precalculus / College Algebra / Trigonometry — Stitz, Zeager · 2013, v3 · EN · CC-BY-NC-SA · §10-11.
Active Calculus — Matt Boelkins · 2024, ed. 2.0 · EN · CC-BY-NC-SA · §8: sequences and series.
Basic Analysis: Introduction to Real Analysis (Vol. I) — Jiří Lebl · 2024, v6.0 · EN · CC-BY-SA · §2: rigorous convergence.

Full catalog at /livros.

Trim 2 Roadmap

Problem styles

Self-assessment

Sources for this lesson