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. sin 75°
963
views
Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
Capitolo 3, Problema 16
Verify each identity. cos² θ (1 + tan² θ) = 1
Guida verificata passo dopo passo1
Recall the Pythagorean identity involving tangent and secant: \(\tan^{2} \theta + 1 = \sec^{2} \theta\).
Rewrite the expression \(\cos^{2} \theta (1 + \tan^{2} \theta)\) by substituting \(1 + \tan^{2} \theta\) with \(\sec^{2} \theta\), so it becomes \(\cos^{2} \theta \cdot \sec^{2} \theta\).
Express \(\sec \theta\) in terms of cosine: \(\sec \theta = \frac{1}{\cos \theta}\), so \(\sec^{2} \theta = \frac{1}{\cos^{2} \theta}\).
Substitute \(\sec^{2} \theta\) back into the expression to get \(\cos^{2} \theta \cdot \frac{1}{\cos^{2} \theta}\).
Simplify the expression by canceling \(\cos^{2} \theta\) in numerator and denominator, which results in \(1\), thus verifying the identity.
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:
2mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Pythagorean Identity
The Pythagorean identity states that sin²θ + cos²θ = 1 for any angle θ. This fundamental relationship between sine and cosine is often used to simplify or verify trigonometric expressions involving squares of sine and cosine.
Video consigliato:
Percorso guidato
6:25
Pythagorean Identities
Definition of Tangent in Terms of Sine and Cosine
Tangent of an angle θ is defined as tan θ = sin θ / cos θ. Understanding this definition allows you to rewrite expressions involving tan²θ in terms of sine and cosine, facilitating simplification and verification of identities.
Video consigliato:
Percorso guidato
5:08Sine, Cosine, & Tangent of 30°, 45°, & 60°
Algebraic Manipulation of Trigonometric Expressions
Verifying identities often requires algebraic skills such as factoring, expanding, and substituting equivalent expressions. Being able to manipulate trigonometric expressions correctly is essential to transform one side of the equation to match the other.
Video consigliato:
Percorso guidato
6:36Simplifying Trig Expressions
Pratica correlata
Domanda del libro di testo
Use a sum or difference formula to find the exact value of each expression. tan 5𝝅/12
875
views
1
rank
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. cos 3x/2 + cos x/2
640
views
Domanda del libro di testo
Find all solutions of each equation. tan x = 0
660
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 x + sin 2x
651
views
Domanda del libro di testo
In Exercises 12–18, solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. 2 sin² x + cos x = 1
470
views