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 87
Capitolo 1, Problema 87

In Exercises 87–92, find the exact value of each expression. Write the answer as a single fraction. Do not use a calculator. sin 𝜋/3 cos 𝜋 - cos 𝜋/3 sin 3𝜋/2

Guida verificata passo dopo passo
1
Recognize that the expression is of the form \(\sin A \cos B - \cos A \sin B\), which matches the sine difference identity: \(\sin(A - B) = \sin A \cos B - \cos A \sin B\).
Identify the angles in the expression: \(A = \frac{\pi}{3}\) and \(B = \frac{2\pi}{3}\), so the expression simplifies to \(\sin\left(\frac{\pi}{3} - \frac{2\pi}{3}\right)\).
Calculate the difference inside the sine function: \(\frac{\pi}{3} - \frac{2\pi}{3} = -\frac{\pi}{3}\).
Use the odd property of sine: \(\sin(-\theta) = -\sin \theta\), so \(\sin\left(-\frac{\pi}{3}\right) = -\sin\left(\frac{\pi}{3}\right)\).
Recall the exact value of \(\sin\left(\frac{\pi}{3}\right)\), which is \(\frac{\sqrt{3}}{2}\), and write the final expression as a single fraction using this value.

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:
6m

Concetti chiave

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

Unit Circle and Special Angles

The unit circle helps determine exact values of sine and cosine for special angles like π/3 and 2π/3. These angles correspond to well-known coordinates on the circle, allowing evaluation without a calculator.
Video consigliato:
Percorso guidato
06:11
Introduction to the Unit Circle

Trigonometric Identities and Angle Multiplication

Understanding how to manipulate trigonometric expressions involving multiples of π is essential. Recognizing that sine and cosine values repeat or change sign at specific intervals aids in simplifying expressions.
Video consigliato:
Percorso guidato
05:06
Double Angle Identities

Algebraic Simplification of Trigonometric Expressions

After substituting exact trigonometric values, combining terms into a single fraction requires careful algebraic manipulation. This includes factoring, common denominators, and simplifying radicals to express the answer neatly.
Video consigliato:
Percorso guidato
6:36
Simplifying Trig Expressions
Pratica correlata
Domanda del libro di testo

Use the circle shown in the rectangular coordinate system to solve Exercises 81–86. Find two angles, in radians, between -2𝜋 and 2𝜋 such that each angle's terminal side passes through the origin and the given point.

E

882
views
Domanda del libro di testo

Use the circle shown in the rectangular coordinate system to solve Exercises 81–86. Find two angles, in radians, between -2𝜋 and 2𝜋 such that each angle's terminal side passes through the origin and the given point.

F

591
views
Domanda del libro di testo

Find the absolute value of the radian measure of the angle that the second hand of a clock moves through in the given time. 3 minutes and 40 seconds

687
views
Domanda del libro di testo

Find the absolute value of the radian measure of the angle that the second hand of a clock moves through in the given time. 55 seconds

655
views
Domanda del libro di testo

Find the exact value of each expression. Write the answer as a single fraction. Do not use a calculator. sin (3𝜋/2) tan (-15𝜋/4) - cos (-5𝜋/3)

708
views
Domanda del libro di testo

Use reference angles to find the exact value of each expression. Do not use a calculator. sin (-17𝜋/3)

748
views