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Lesson 12 — Trigonometric circle and radians

Generalization of trigonometric ratios via the unit circle. Radians as the natural unit. Fundamental identities and periodicity.

Used in: 1st year HS

P(\theta) = (\cos\theta, \sin\theta) \quad \text{on the unit circle}

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Rigorous notation, full derivation, hypotheses

Definition via the unit circle

Radians vs degrees

(1)

what this means · One radian is the central angle that subtends an arc of length equal to the radius. Since the perimeter of the unit circle is 2π, a full turn (360°) equals 2π radians. In calculus, radians are always used: identities such as (sin x)' = cos x only hold in this unit.

Trigonometric circle. For each angle θ, the point P(θ) = (cos θ, sin θ). Periodicity: rotating by 2π returns to the starting point.

Quadrant	$\theta$	$\sin\theta$	$\cos\theta$	$\tan\theta$
I	$(0, \pi/2)$	+	+	+
II	$(\pi/2, \pi)$	+	−	−
III	$(\pi, 3\pi/2)$	−	−	+
IV	$(3\pi/2, 2\pi)$	−	+	−

Special angles

$\theta$	$0$	$\pi/6$	$\pi/4$	$\pi/3$	$\pi/2$	$\pi$	$3\pi/2$	$2\pi$
$\sin$	$0$	$1/2$	$\sqrt 2/2$	$\sqrt 3/2$	$1$	$0$	$-1$	$0$
$\cos$	$1$	$\sqrt 3/2$	$\sqrt 2/2$	$1/2$	$0$	$-1$	$0$	$1$

Fundamental identities

$\sin(-\theta) = -\sin\theta \quad (\sin \text{ is odd})$ $\cos(-\theta) = \cos\theta \quad (\cos \text{ is even})$ $\sin(\theta + \pi/2) = \cos\theta, \quad \cos(\theta + \pi/2) = -\sin\theta$

Exercise list

40 exercises · 10 with worked solution (25%)

Application 20Understanding 10Modeling 9Challenge 1

Ex. 12.1Application
Convert $60°$ to radians.
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Ex. 12.2Application
Convert $225°$ to radians.
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Ex. 12.3Application
Convert $120°$ to radians.
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Ex. 12.4Application
Convert $\pi/3$ rad to degrees.
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Ex. 12.5ApplicationAnswer key
Convert $7\pi/4$ rad to degrees.
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Ex. 12.6ApplicationAnswer key
Convert $1$ rad to degrees (approximately).
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Ex. 12.7Application
Convert $90°$ to radians.
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Ex. 12.8Application
Convert $-150°$ to radians.
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Ex. 12.9ApplicationAnswer key
Convert $\pi/12$ rad to degrees.
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Ex. 12.10Application
Convert $400°$ to radians.
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Ex. 12.11ApplicationAnswer key
Compute $\sin(\pi/6)$ and $\cos(\pi/6)$ .
Solve onlineref: OpenStax A&T §8.3
Ex. 12.12ApplicationAnswer key
Compute $\sin(2\pi/3)$ and $\cos(2\pi/3)$ .
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Ex. 12.13Application
Compute $\sin(\pi)$ and $\cos(\pi)$ .
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Ex. 12.14Application
Compute $\sin(3\pi/2)$ and $\cos(3\pi/2)$ .
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Ex. 12.15Application
Compute $\sin(7\pi/6)$ and $\cos(7\pi/6)$ .
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Ex. 12.16Application
Compute $\sin(11\pi/6)$ .
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Ex. 12.17Application
Compute $\cos(5\pi/4)$ .
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Ex. 12.18Application
Compute $\sin(5\pi/3)$ .
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Ex. 12.19Application
Compute $\tan(\pi/3)$ .
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Ex. 12.20Application
Compute $\tan(7\pi/6)$ .
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Ex. 12.21Understanding
Verify $\sin^2(\pi/3) + \cos^2(\pi/3) = 1$ .
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Ex. 12.22Understanding
Show that $\sin(-\pi/4) = -\sin(\pi/4)$ .
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Ex. 12.23UnderstandingAnswer key
Show that $\cos(-\pi/3) = \cos(\pi/3)$ .
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Ex. 12.24Understanding
Compute $\sin(105°) = \sin(60° + 45°)$ using the sum formula.
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Ex. 12.25UnderstandingAnswer key
Compute $\cos(15°) = \cos(45° - 30°)$ .
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Ex. 12.26Understanding
Compute $\sin(2 \cdot 30°) = 2\sin 30° \cos 30°$ . Verify it equals $\sin 60°$ .
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Ex. 12.27Understanding
Show that $\cos(2\theta) = 1 - 2\sin^2\theta$ .
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Ex. 12.28Understanding
Show that $\sin(\theta + \pi) = -\sin\theta$ .
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Ex. 12.29Understanding
Determine $\theta \in [0, 2\pi)$ such that $\sin\theta = \cos\theta$ .
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Ex. 12.30Understanding
In which quadrant is $\sin\theta > 0$ and $\cos\theta < 0$ ?
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Ex. 12.31ModelingAnswer key
A disc spins at 33 rotations per minute. Compute the angular velocity in rad/s.
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Ex. 12.32Modeling
A pendulum traces an arc of 30°. Converting to rad: what is the arc length if the string is 1.5 m?
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Ex. 12.33Modeling
An analog clock: the minute hand rotates 360° per hour. How many rad/min?
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Ex. 12.34Modeling
The Earth rotates 360° in 24 h. Angular velocity in rad/h?
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Ex. 12.35Modeling
In DSP, a sinusoidal wave $y(t) = \sin(\omega t)$ has $\omega = 2\pi f$ . For $f = 60$ Hz (Brazilian power grid), what is $\omega$ ?
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Ex. 12.36ModelingAnswer key
A motor spins at 1,800 rpm. Angular velocity in rad/s?
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Ex. 12.37Modeling
A bike wheel has radius 35 cm. If the linear speed is 20 km/h, what is the angular velocity in rad/s?
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Ex. 12.38Modeling
In GPS, satellites orbit Earth at ~14,000 km/h on a circular orbit of radius 26,600 km. Angular velocity?
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Ex. 12.39Modeling
In mechanics, the phase angle of an oscillator is $\theta(t) = \omega t + \phi$ . For $\omega = 2\pi$ rad/s, $\phi = \pi/4$ , what are $\theta(0)$ and $\theta(1)$ ?
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Ex. 12.40ChallengeAnswer key
A rotating machine is in equilibrium if $\sum F_i \cos\theta_i = 0$ . Verify for 3 equal forces $120°$ apart.
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Sources for this lesson

Algebra and Trigonometry — Jay Abramson et al. (OpenStax) · 2022, 2nd ed · EN · CC-BY · §8.3-8.5: unit circle, fundamental identities, sum/difference formulas. Primary source.
Precalculus / College Algebra / Trigonometry — Stitz, Zeager · 2013, v3 · EN · CC-BY-NC-SA · §10.3-10.4: unit circle and identities. Source for block C.
College Trigonometry — Stitz, Zeager · 2013 · EN · CC-BY-NC-SA · ch. 10: extensive treatment.
Geometria e Trigonometria — Wikilivros · live · PT-BR · CC-BY-SA · Portuguese-language reference.