Skip to main content
Ch. 2 - Acute Angles and Right Triangles
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 2 - Acute Angles and Right TrianglesProblema 39
Capitolo 3, Problema 39

Find one solution for each equation. Assume all angles involved are acute angles. See Example 3. sec(3β + 10°) = csc(β + 8°)

Guida verificata passo dopo passo
1
Recall the definitions of the secant and cosecant functions in terms of sine and cosine: \(\sec \theta = \frac{1}{\cos \theta}\) and \(\csc \theta = \frac{1}{\sin \theta}\).
Rewrite the given equation \(\sec(3\beta + 10^\circ) = \csc(\beta + 8^\circ)\) using these definitions: \(\frac{1}{\cos(3\beta + 10^\circ)} = \frac{1}{\sin(\beta + 8^\circ)}\).
Cross-multiply to get an equation involving sine and cosine: \(\sin(\beta + 8^\circ) = \cos(3\beta + 10^\circ)\).
Use the co-function identity \(\cos \theta = \sin(90^\circ - \theta)\) to rewrite the right side: \(\sin(\beta + 8^\circ) = \sin(90^\circ - (3\beta + 10^\circ))\).
Simplify the right side inside the sine function and then solve the resulting equation \(\sin A = \sin B\) for \(\beta\), considering that \(\beta\) is an acute angle (between \(0^\circ\) and \(90^\circ\)). Remember that \(\sin A = \sin B\) implies \(A = B + 360^\circ k\) or \(A = 180^\circ - B + 360^\circ k\) for any integer \(k\).

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.

Reciprocal Trigonometric Functions

Secant (sec) and cosecant (csc) are reciprocal functions of cosine and sine, respectively. Specifically, sec(θ) = 1/cos(θ) and csc(θ) = 1/sin(θ). Understanding these relationships allows rewriting the equation in terms of sine and cosine for easier manipulation.
Video consigliato:
Percorso guidato
6:04
Introduction to Trigonometric Functions

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding angle values that satisfy the equation within the given domain. Since the problem restricts angles to acute values, solutions must be between 0° and 90°, which limits possible solutions.
Video consigliato:
Percorso guidato
4:34
How to Solve Linear Trigonometric Equations

Angle Sum and Multiple Angle Arguments

The equation involves expressions like 3β + 10° and β + 8°, which are linear combinations of the variable β. Understanding how to handle these composite angles is essential, as it requires applying algebraic techniques and possibly inverse trigonometric functions to isolate β.
Video consigliato:
Percorso guidato
04:46
Coterminal Angles
Pratica correlata
Domanda del libro di testo

Find one solution for each equation. Assume all angles involved are acute angles. See Example 3. cos(2θ + 50°) = sin(2θ - 20°)

585
views
Domanda del libro di testo

Determine whether each statement is true or false. See Example 4. tan 28° ≤ tan 40°

575
views
Domanda del libro di testo

Find the exact value of each expression. See Example 3. sin 1305°

635
views
Domanda del libro di testo

Find one solution for each equation. Assume all angles involved are acute angles. See Example 3. csc(β + 40°) = sec(β - 20°)

618
views
Domanda del libro di testo

Find the exact value of each expression. See Example 3. tan(-1020°)

597
views
Domanda del libro di testo

Solve each right triangle. In each case, C = 90°. If angle information is given in degrees and minutes, give answers in the same way. If angle information is given in decimal degrees, do likewise in answers. When two sides are given, give angles in degrees and minutes. See Examples 1 and 2. B = 39°09', c = 0.6231 m

579
views