Skip to main content
Ch. 1 - Angles and the Trigonometric Functions
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 3rd EditionCh. 1 - Angles and the Trigonometric FunctionsProblema 1.2.63
Capitolo 1, Problema 1.2.63

In Exercises 63–68, find the exact value of each expression. Do not use a calculator. tan(𝜋/3)/2 - 1/sec(𝜋/6)

Guida verificata passo dopo passo
1
Identify the given expression: \(\frac{\tan \frac{\pi}{3} - 1}{2 \sec \frac{\pi}{6}}\).
Recall the exact values of the trigonometric functions involved: \(\tan \frac{\pi}{3} = \sqrt{3}\), and \(\sec \frac{\pi}{6} = \frac{1}{\cos \frac{\pi}{6}}\).
Find \(\cos \frac{\pi}{6}\) using the known exact value: \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\), so \(\sec \frac{\pi}{6} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}\).
Substitute these values back into the expression: \(\frac{\sqrt{3} - 1}{2 \times \frac{2}{\sqrt{3}}}\).
Simplify the denominator and then the entire fraction step-by-step, rationalizing denominators if necessary, to find the exact value.

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:
5m

Concetti chiave

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

Exact Values of Special Angles

Certain angles like π/3 and π/6 have well-known exact trigonometric values derived from special triangles. For example, tan(π/3) equals √3, and sec(π/6) equals 2/√3. Knowing these values allows precise calculation without a calculator.
Video consigliato:
Percorso guidato
04:39
45-45-90 Triangles

Definition and Relationship of Trigonometric Functions

Understanding how tangent and secant relate to sine and cosine is essential. Tangent is the ratio of sine to cosine (tan θ = sin θ / cos θ), while secant is the reciprocal of cosine (sec θ = 1 / cos θ). This helps in simplifying and evaluating expressions.
Video consigliato:
Percorso guidato
6:04
Introduction to Trigonometric Functions

Simplifying Trigonometric Expressions

Combining and simplifying expressions involving trigonometric functions requires algebraic manipulation, such as finding common denominators or rationalizing denominators. This skill ensures the exact value is expressed in simplest form without approximations.
Video consigliato:
Percorso guidato
6:36
Simplifying Trig Expressions
Pratica correlata
Domanda del libro di testo

In Exercises 57–70, find a positive angle less than or that is coterminal with the given angle. -760°

642
views
Domanda del libro di testo

In Exercises 63–68, find the exact value of each expression. Do not use a calculator. 1 + sin² 40° + sin² 50°

601
views
Domanda del libro di testo

In Exercises 35–60, find the reference angle for each angle. 17𝜋 / 6

628
views
Domanda del libro di testo

In Exercises 41–56, use the circle shown in the rectangular coordinate system to draw each angle in standard position. State the quadrant in which the angle lies. When an angle's measure is given in radians, work the exercise without converting to degrees.

120°

635
views
Domanda del libro di testo

In Exercises 57–70, find a positive angle less than or that is coterminal with the given angle. 395°

679
views
Domanda del libro di testo

In Exercises 71–74, find the length of the arc on a circle of radius r intercepted by a central angle θ. Express arc length in terms of 𝜋. Then round your answer to two decimal places. Radius, r: 12 inches Central Angle, θ: θ = 45°

702
views