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Lesson 13 — Trigonometric functions

Graphs of sin, cos, tan. Periodicity, amplitude, phase, frequency. Modeling periodic phenomena.

Used in: 1st year HS · Physics (waves) · Engineering (signals)

y(t) = A \sin(\omega t + \varphi) + k

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Rigorous notation, full derivation, hypotheses

Definition and parameters

Generalized sinusoidal function

For $y(t) = A \sin(\omega t + \varphi) + k$ with $A, \omega > 0$ :

Amplitude $A$ : distance from the midline to the peaks. Range: $[k - A, k + A]$ .
Angular frequency $\omega$ : speed of oscillation. Period $T = 2\pi/\omega$ . Frequency $f = 1/T = \omega/(2\pi)$ .
Initial phase $\varphi$ : horizontal shift. $y$ reaches its maximum when $\omega t + \varphi = \pi/2$ , that is, $t = (\pi/2 - \varphi)/\omega$ .
Vertical shift $k$ : middle of the graph.

Graphs of sin x (blue) and cos x (orange). Phase-shifted by π/2. Both have amplitude 1 and period 2π.

Exercise list

40 exercises · 10 with worked solution (25%)

Application 20Understanding 4Modeling 12Challenge 2Proof 2

Ex. 13.1Application
Sketch $y = 2\sin x$ on the interval $[0, 2\pi]$ . Identify amplitude and period.
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Ex. 13.2Application
Sketch $y = \sin(2x)$ . Period?
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Ex. 13.3Application
Sketch $y = \cos(x/2)$ . Period?
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Ex. 13.4Application
Sketch $y = \sin(x - \pi/4)$ . Phase shift?
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Ex. 13.5Application
Sketch $y = 3\sin x + 1$ . Identify range.
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Ex. 13.6Application
Identify amplitude, period, phase in $y = 4\sin(3x - \pi)$ .
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Ex. 13.7ApplicationAnswer key
Identify the range of $y = 2\cos(x) - 1$ .
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Ex. 13.8Application
For $y = \sin(\pi t)$ , what is the period in seconds?
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Ex. 13.9Application
For $y = \cos(2\pi t/T)$ , show that the period is $T$ .
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Ex. 13.10ApplicationAnswer key
Sketch $y = \tan x$ on $(-\pi/2, \pi/2)$ .
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Ex. 13.11Application
Solve $\sin x = 1/2$ on $[0, 2\pi)$ .
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Ex. 13.12Application
Solve $\cos x = 0$ on $[0, 2\pi)$ .
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Ex. 13.13Application
Solve $\tan x = 1$ on $[0, 2\pi)$ .
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Ex. 13.14Application
Solve $\sin x = -\sqrt 2/2$ on $[0, 2\pi)$ .
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Ex. 13.15Application
Solve $\cos(2x) = 1/2$ on $[0, 2\pi)$ .
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Ex. 13.16Application
Solve $\sin x = \cos x$ on $[0, 2\pi)$ .
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Ex. 13.17ApplicationAnswer key
Solve $2\sin x - 1 = 0$ on $[0, 2\pi)$ .
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Ex. 13.18Application
Solve $\sin^2 x = 1/4$ on $[0, 2\pi)$ .
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Ex. 13.19Application
Solve $\sin(x + \pi/3) = 1/2$ on $[0, 2\pi)$ .
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Ex. 13.20ApplicationAnswer key
Solve $\tan(2x) = \sqrt{3}$ on $[0, 2\pi)$ .
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Ex. 13.21Modeling
The tide in Salvador oscillates between 0.5 m and 2.5 m with a period of 12 h. Model the height $h(t)$ as a function of time.
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Ex. 13.22ModelingAnswer key
Brazilian power grid voltage: $V(t) = 311 \sin(120\pi t)$ . RMS voltage?
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Ex. 13.23Modeling
The height of a Ferris wheel seat (radius 10 m, axle 12 m above the ground) makes 1 turn every 4 min. Model $h(t)$ .
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Ex. 13.24ModelingAnswer key
A pure 440 Hz tone has $p(t) = A \sin(880\pi t)$ . How many oscillations in 1 second?
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Ex. 13.25ModelingAnswer key
The monthly average temperature in Brasília oscillates between 18°C (July) and 23°C (October). Model $T(m)$ with $m$ in months.
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Ex. 13.26Modeling
A 1 m pendulum oscillates at $\omega = \sqrt{g/L}$ . For $g = 9{,}81$ , what is the period?
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Ex. 13.27Modeling
Mass-spring system: $m = 0{,}5$ kg, $k = 50$ N/m. Natural frequency $\omega = \sqrt{k/m}$ . Compute it.
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Ex. 13.28Modeling
In vibration mechanics, $x(t) = 5 \sin(2\pi t)$ cm. Maximum velocity?
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Ex. 13.29Modeling
Tides under lunar influence only: period $T = 12$ h $25$ min. Frequency?
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Ex. 13.30Modeling
A Cepheid star varies in brightness with $T = 5{,}4$ days and amplitude 0.8 magnitudes. Model $m(t)$ .
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Ex. 13.31Modeling
A submarine's diving depth oscillates as $d(t) = 100 + 30 \sin(\pi t / 30)$ m. Maximum and minimum depth? Period?
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Ex. 13.32ModelingAnswer key
In GPS, carrier signal of 1,575 MHz. Period in seconds? (GPS waves are nearly instantaneous — hence the precision.)
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Ex. 13.33Understanding
Show that the sum $\sin x + \sin(x + 2\pi/3) + \sin(x + 4\pi/3) = 0$ for every $x$ . (This is the result behind three-phase motors.)
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Ex. 13.34UnderstandingAnswer key
Sketch $y = \sin x + \sin 3x$ . (Lessons from Fourier — adding harmonics.)
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Ex. 13.35Understanding
Show that $A \sin(\omega t) + B \cos(\omega t) = \sqrt{A^2 + B^2} \sin(\omega t + \varphi)$ with $\tan\varphi = B/A$ .
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Ex. 13.36Understanding
Verify: $\sin(x) + \sin(x + \pi) = 0$ .
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Ex. 13.37Challenge
Solve $\sin^2 x - 3\sin x + 2 = 0$ on $[0, 2\pi)$ . (Use $u = \sin x$ .)
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Ex. 13.38ChallengeAnswer key
Solve $2\sin^2 x + \sin x - 1 = 0$ on $[0, 2\pi)$ .
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Ex. 13.39Proof
Prove $\cos(2x) = 1 - 2\sin^2 x$ .
Ex. 13.40Proof
Prove that $\sin(x + 2\pi) = \sin x$ .

Sources for this lesson

Algebra and Trigonometry — Jay Abramson et al. (OpenStax) · 2022, 2nd ed · EN · CC-BY · §8.4-8.6: graphs, transformations, modeling. Primary source.
Precalculus / College Algebra / Trigonometry — Stitz, Zeager · 2013, v3 · EN · CC-BY-NC-SA · §11: graphs and identities.
University Physics (Volume 1) — OpenStax · 2016 · EN · CC-BY · ch. 16: waves and oscillations.