Find sinθ.tan θ = -(√7)/2, sec θ \u003e 0

Ch. 5 - Trigonometric Identities

Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo

Tutti i libri di testo

Lial 12th Edition

Ch. 5 - Trigonometric Identities

Problema 5.1.18

Capitolo 6, Problema 5.1.18

Find sinθ.
tan θ = -(√7)/2, sec θ > 0

Guida verificata passo dopo passo

Identify the given information: \( \tan \theta = -\frac{\sqrt{7}}{2} \) and \( \sec \theta > 0 \). Recall that \( \sec \theta = \frac{1}{\cos \theta} \), so \( \sec \theta > 0 \) means \( \cos \theta > 0 \).

Determine the quadrant where \( \theta \) lies. Since \( \tan \theta \) is negative and \( \cos \theta \) is positive, \( \theta \) must be in the fourth quadrant (where cosine is positive and tangent is negative).

Use the identity relating tangent and sine and cosine: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). Let \( \cos \theta = x \), then \( \sin \theta = \tan \theta \times x = -\frac{\sqrt{7}}{2} x \).

Apply the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). Substitute \( \sin \theta = -\frac{\sqrt{7}}{2} x \) and \( \cos \theta = x \) to get \( \left(-\frac{\sqrt{7}}{2} x\right)^2 + x^2 = 1 \).

Solve the equation for \( x \) (which is \( \cos \theta \)), then use \( \sin \theta = -\frac{\sqrt{7}}{2} x \) to find \( \sin \theta \). Remember to choose the sign of \( \sin \theta \) consistent with the quadrant (fourth quadrant means \( \sin \theta < 0 \)).

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Concetti chiave

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

Trigonometric Ratios and Their Relationships

Trigonometric ratios like sine, cosine, and tangent relate the angles of a right triangle to the ratios of its sides. Knowing one ratio, such as tangent, allows you to find others using identities or the Pythagorean theorem.

Video consigliato:

Percorso guidato

6:04

Introduction to Trigonometric Functions

Sign of Trigonometric Functions in Different Quadrants

The signs of sine, cosine, and tangent depend on the quadrant where the angle lies. Given tan θ = -(√7)/2 and sec θ > 0, you can determine the quadrant by recalling that sec θ is positive where cosine is positive.

Video consigliato:

Percorso guidato

6:04

Introduction to Trigonometric Functions

Using Pythagorean Identities to Find Missing Ratios

Pythagorean identities like 1 + tan²θ = sec²θ help find unknown trigonometric values. By substituting the given tangent value and using the sign information, you can calculate sine θ accurately.

Video consigliato:

Percorso guidato

6:25

Pythagorean Identities

Pratica correlata

Domanda del libro di testo

Match each expression in Column I with its value in Column II.

(2 tan 15°)/(1 - tan² 15°)

views

Domanda del libro di testo

Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.

csc² t - 1

721

views

Domanda del libro di testo

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

(sec θ - 1) (sec θ + 1)

777

views

Domanda del libro di testo

Verify that each equation is an identity.

(tan² α + 1)/ sec α = sec α

797

views

Domanda del libro di testo

Verify that each equation is an identity.

[(sec θ - tan θ)² + 1]/(sec θ csc θ - tan θ csc θ) = 2 tan θ

877