Domanda del libro di testo
In Exercises 7–12, find the radian measure of the central angle of a circle of radius r that intercepts an arc of length s. Radius, r: 1 meter Arc Length, s: 600 centimeters
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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Capitolo 1, Problema 12
In Exercises 9–16, evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined. csc 𝜋
Guida verificata passo dopo passo1
Recall that the cosecant function is the reciprocal of the sine function, so \(\csc \theta = \frac{1}{\sin \theta}\).
Identify the angle given: \(\pi\) radians, which is a quadrantal angle located on the negative x-axis of the unit circle.
Evaluate \(\sin \pi\). Since \(\sin \pi = 0\), substitute this value into the reciprocal expression for cosecant.
Since \(\csc \pi = \frac{1}{\sin \pi} = \frac{1}{0}\), recognize that division by zero is undefined.
Conclude that \(\csc \pi\) is undefined because the sine of \(\pi\) is zero, making the reciprocal undefined.
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Quadrantal Angles
Quadrantal angles are angles that lie on the x- or y-axis in the unit circle, typically multiples of π/2 (e.g., 0, π/2, π, 3π/2). These angles have special sine and cosine values, often 0, ±1, which affect the evaluation of trigonometric functions.
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Percorso guidato
6:36
Quadratic Formula
Cosecant Function (csc)
The cosecant function is the reciprocal of the sine function, defined as csc(θ) = 1/sin(θ). It is undefined wherever sin(θ) = 0, which commonly occurs at quadrantal angles like 0, π, and 2π.
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6:22Graphs of Secant and Cosecant Functions
Evaluating Trigonometric Functions at Specific Angles
To evaluate trigonometric functions at specific angles, substitute the angle into the function and use known sine and cosine values from the unit circle. For quadrantal angles, check if the function is defined or undefined due to division by zero.
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3:48Evaluate Composite Functions - Special Cases
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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.
sec 11𝜋/6
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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.
In Exercises 11–18, continue to refer to the figure at the bottom of the previous page. csc 4𝜋/3
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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°
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Domanda del libro di testo
In Exercises 8–12, draw each angle in standard position. -135°
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Domanda del libro di testo
In Exercises 8–13, find the exact value of each expression. Do not use a calculator. cot (-8𝜋/3)
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