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 11a

Capitolo 3, Problema 11a

Use the given information to find the exact value of each of the following: sin 2θ
cot θ = 2, θ lies in quadrant III.

Guida verificata passo dopo passo

Recall the double-angle identity for sine: \(\sin 2\theta = 2 \sin \theta \cos \theta\).

Use the given information \(\cot \theta = 2\) to express \(\tan \theta\) as \(\frac{1}{2}\), since \(\cot \theta = \frac{1}{\tan \theta}\).

Determine the signs of \(\sin \theta\) and \(\cos \theta\) in quadrant III, where both sine and cosine are negative.

Use the identity \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) and the value \(\tan \theta = \frac{1}{2}\) to set up a ratio between \(\sin \theta\) and \(\cos \theta\).

Find exact values for \(\sin \theta\) and \(\cos \theta\) using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\), then substitute these into the double-angle formula \(\sin 2\theta = 2 \sin \theta \cos \theta\).

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.

Cotangent and Its Relationship to Sine and Cosine

Cotangent (cot θ) is the reciprocal of tangent, defined as cot θ = cos θ / sin θ. Knowing cot θ allows us to express sine and cosine in terms of each other, which is essential for finding sin 2θ. Understanding this relationship helps in determining the values of sine and cosine from cotangent.

Video consigliato:

Percorso guidato

5:08

Sine, Cosine, & Tangent of 30°, 45°, & 60°

Sign of Trigonometric Functions in Quadrants

The sign of sine, cosine, and tangent functions depends on the quadrant where the angle lies. Since θ is in quadrant III, both sine and cosine are negative, but tangent (and cotangent) is positive. This information is crucial to assign correct signs to sine and cosine values when calculating sin 2θ.

Video consigliato:

Percorso guidato

6:36

Quadratic Formula

Double-Angle Identity for Sine

The double-angle identity for sine states that sin 2θ = 2 sin θ cos θ. This formula allows us to find sin 2θ once sin θ and cos θ are known or expressed in terms of cot θ. Applying this identity is the key step in solving the problem.

Video consigliato:

Percorso guidato

05:06

Double Angle Identities

Pratica correlata

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 7x ﹣ sin 3x

647

views

Domanda del libro di testo

Find all solutions of each equation. sin x = (√3)/2

524

views

Domanda del libro di testo

Use the given information to find the exact value of each of the following: tan 2θ

cot θ = 2, θ lies in quadrant III.

781

views

Domanda del libro di testo