Ch. 3 - Trigonometric Identities and Equations

Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo

Blitzer 3rd Edition

Ch. 3 - Trigonometric Identities and Equations

Problema 19

Capitolo 3, Problema 19

Write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. 2cos² 𝝅/8﹣ 1

Guida verificata passo dopo passo

Recognize that the given expression is of the form \(\cos^2 x - 1\), where \(x = \frac{\pi}{8}\). Recall the double-angle identity for cosine: \(\cos 2x = 2\cos^2 x - 1\).

Rearrange the double-angle identity to express \(\cos^2 x - 1\) in terms of \(\cos 2x\): starting from \(\cos 2x = 2\cos^2 x - 1\), subtract 1 from both sides and then isolate \(\cos^2 x - 1\).

Express \(\cos^2 x - 1\) as \(\frac{\cos 2x - 1}{2}\) by manipulating the identity: \(\cos 2x = 2\cos^2 x - 1 \Rightarrow 2\cos^2 x = \cos 2x + 1 \Rightarrow \cos^2 x = \frac{\cos 2x + 1}{2}\), so \(\cos^2 x - 1 = \frac{\cos 2x + 1}{2} - 1\).

Simplify the expression \(\cos^2 x - 1\) to \(\frac{\cos 2x - 1}{2}\), which is now written as a cosine double-angle expression.

Substitute \(x = \frac{\pi}{8}\) into the double-angle expression to get \(\frac{\cos \left(2 \times \frac{\pi}{8}\right) - 1}{2} = \frac{\cos \frac{\pi}{4} - 1}{2}\). Then, use the exact value of \(\cos \frac{\pi}{4}\) to find the exact value of the original expression.

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:

Concetti chiave

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

Double-Angle Identities

Double-angle identities express trigonometric functions of twice an angle in terms of functions of the original angle. For example, cos(2θ) = 2cos²θ - 1 or cos(2θ) = 1 - 2sin²θ. These identities help rewrite expressions like cos²(θ) in terms of cos(2θ), simplifying evaluation.

Video consigliato:

Percorso guidato

05:06

Double Angle Identities

Exact Values of Trigonometric Functions

Exact values refer to precise trigonometric values for special angles, often expressed in terms of square roots and fractions. Angles like π/8 or π/4 have known exact sine, cosine, and tangent values, which are essential for finding exact results without decimal approximations.

Video consigliato:

Percorso guidato

6:04

Introduction to Trigonometric Functions

Trigonometric Expression Simplification

Simplifying trigonometric expressions involves rewriting them using identities to reduce complexity. This includes converting powers of sine or cosine into single trigonometric functions of multiple angles, enabling easier evaluation and clearer understanding of the expression.

Video consigliato:

Percorso guidato

6:36

Simplifying Trig Expressions

Pratica correlata

Domanda del libro di testo

Find all solutions of each equation. 2 cos x + √ 3 = 0

521

views

Domanda del libro di testo

Be sure that you've familiarized yourself with the second set of formulas presented in this section by working C5–C8 in the Concept and Vocabulary Check. In Exercises 9–22, express each sum or difference as a product. If possible, find this product's exact value. sin 75° + sin 15°

677

views

Domanda del libro di testo

Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find the exact value of each expression. cos(135° + 30°)

904

views

Domanda del libro di testo

Find all solutions of each equation. 4 sin θ﹣1 = 2 sin θ

470

views

Domanda del libro di testo

Write each expression as the sine, cosine, or tangent of a double angle. Then find the exact value of the expression. cos² 105° - sin² 105°

769

views

Domanda del libro di testo

677

views