Domanda del libro di testo
In Exercises 8–13, find the exact value of each expression. Do not use a calculator. sec 22𝜋 3
680
views
Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
Capitolo 1, Problema 12
In Exercises 8–13, find the exact value of each expression. Do not use a calculator. cot (-8𝜋/3)
Guida verificata passo dopo passo1
Recognize that the cotangent function is periodic with period \(\pi\), so we can simplify the angle \(-\frac{8\pi}{3}\) by adding or subtracting multiples of \(\pi\) to find a coterminal angle within a standard interval, such as \([0, 2\pi)\).
Add \(3\pi\) (which is \(\pi\) times 3) to \(-\frac{8\pi}{3}\) to find a positive coterminal angle: \(-\frac{8\pi}{3} + 3\pi = -\frac{8\pi}{3} + \frac{9\pi}{3} = \frac{\pi}{3}\).
Now, evaluate \(\cot\left(\frac{\pi}{3}\right)\). Recall that \(\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\).
Use the known exact values for sine and cosine at \(\frac{\pi}{3}\): \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\) and \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\).
Substitute these values into the cotangent formula: \(\cot\left(\frac{\pi}{3}\right) = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}\), and simplify the fraction 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:
6mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Cotangent Function
Cotangent is a trigonometric function defined as the ratio of the cosine to the sine of an angle, cot(θ) = cos(θ)/sin(θ). It is the reciprocal of the tangent function and is periodic with period π. Understanding cotangent helps in evaluating expressions involving cot(θ) without a calculator.
Video consigliato:
Percorso guidato
5:37
Introduction to Cotangent Graph
Angle Reduction Using Coterminal Angles
Angles differing by full rotations (multiples of 2π) are coterminal and have the same trigonometric values. To simplify cot(-8π/3), add or subtract multiples of 2π to find an equivalent angle within the standard interval [0, 2π) or (-π, π]. This step is crucial for evaluating trigonometric functions exactly.
Video consigliato:
Percorso guidato
04:46Coterminal Angles
Reference Angles and Sign Determination
After reducing the angle to a standard position, identify its reference angle in the first quadrant to find exact trigonometric values. Also, determine the sign of the function based on the quadrant where the angle lies, using the ASTC (All Students Take Calculus) rule. This ensures the correct exact value of cotangent.
Video consigliato:
Percorso guidato
5:31Reference Angles on the Unit Circle
Pratica correlata
Domanda del libro di testo
In Exercises 5–18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of
0, 𝜋, 𝜋, 𝜋, 2𝜋, 5𝜋, 𝜋, 7𝜋, 4𝜋, 3𝜋, 5𝜋, 11𝜋, and 2𝜋.
6 3 2 3 6 6 3 2 3 6
Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.
<IMAGE>
In Exercises 11–18, continue to refer to the figure at the bottom of the previous page.
csc 7𝜋/6
414
views
Domanda del libro di testo
In Exercises 9–16, evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined. csc 𝜋
578
views
Domanda del libro di testo
Use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.
<IMAGE>
csc 45°
504
views
Domanda del libro di testo
In Exercises 8–12, draw each angle in standard position. -135°
798
views
Domanda del libro di testo
Use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.
<IMAGE>
sec 45°
550
views