Skip to main content
Ch. 1 - Angles and the Trigonometric Functions
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 3rd EditionCh. 1 - Angles and the Trigonometric FunctionsProblema 1.RE.58
Capitolo 1, Problema 1.RE.58

In Exercises 49–59, find the exact value of each expression. Do not use a calculator. sin (22𝜋/3)

Guida verificata passo dopo passo
1
Recognize that the angle given is in radians: \(\frac{22\pi}{3}\). Since the sine function is periodic with period \(2\pi\), we can reduce the angle by subtracting multiples of \(2\pi\) to find a coterminal angle within the standard interval \([0, 2\pi)\).
Calculate how many full \(2\pi\) rotations fit into \(\frac{22\pi}{3}\). Since \(2\pi = \frac{6\pi}{3}\), divide \(\frac{22\pi}{3}\) by \(\frac{6\pi}{3}\) to find the quotient: \(\frac{22\pi/3}{6\pi/3} = \frac{22}{6} = 3 + \frac{4}{6} = 3 + \frac{2}{3}\).
Subtract \(3 \times 2\pi = 6\pi\) from \(\frac{22\pi}{3}\) to find the coterminal angle: \(\frac{22\pi}{3} - 6\pi = \frac{22\pi}{3} - \frac{18\pi}{3} = \frac{4\pi}{3}\).
Now, evaluate \(\sin \left( \frac{4\pi}{3} \right)\). Recall that \(\frac{4\pi}{3}\) is in the third quadrant where sine is negative, and it corresponds to an angle of \(\pi + \frac{\pi}{3}\).
Use the sine addition formula or reference the unit circle to express \(\sin \left( \frac{4\pi}{3} \right)\) as \(-\sin \left( \frac{\pi}{3} \right)\), and recall that \(\sin \left( \frac{\pi}{3} \right) = \frac{\sqrt{3}}{2}\).

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
5m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Radian Measure and Angle Conversion

Radians are a way to measure angles based on the radius of a circle, where 2π radians equal 360 degrees. Understanding how to convert between radians and degrees or simplify radian measures is essential for evaluating trigonometric functions accurately.
Video consigliato:
Percorso guidato
5:04
Converting between Degrees & Radians

Periodic Properties of the Sine Function

The sine function is periodic with a period of 2π, meaning sin(θ) = sin(θ + 2πk) for any integer k. This property allows simplification of angles outside the standard interval [0, 2π) to equivalent angles within this range, facilitating exact value determination.
Video consigliato:
Percorso guidato
5:33
Period of Sine and Cosine Functions

Exact Values of Sine for Special Angles

Certain angles, such as π/6, π/4, and π/3, have known exact sine values involving simple fractions and square roots. Decomposing or reducing complex angles into sums or differences of these special angles helps find exact sine values without a calculator.
Video consigliato:
Percorso guidato
3:28
Common Trig Functions For 45-45-90 Triangles
Pratica correlata
Domanda del libro di testo

In Exercises 49–59, find the exact value of each expression. Do not use a calculator. tan 120°

716
views
Domanda del libro di testo

In Exercises 49–59, find the exact value of each expression. Do not use a calculator. csc(-2𝜋/3)

648
views
Domanda del libro di testo

In Exercises 49–59, find the exact value of each expression. Do not use a calculator. cos (11𝜋 / 6)

707
views
Domanda del libro di testo

In Exercises 49–59, find the exact value of each expression. Do not use a calculator. cos(-35𝜋 / 6)

635
views
Domanda del libro di testo

In Exercises 49–59, find the exact value of each expression. Do not use a calculator. sin 240°

729
views
Domanda del libro di testo

In Exercises 49–59, find the exact value of each expression. Do not use a calculator. cot(-210°)

677
views