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 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
Guida verificata passo dopo passo1
Identify the angle given: \(t = \frac{4\pi}{3}\). Locate this angle on the unit circle diagram.
Find the coordinates corresponding to \(t = \frac{4\pi}{3}\) on the unit circle. From the image, the coordinates are \(\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\).
Recall that for any angle \(t\) on the unit circle, \(\sin t\) is the y-coordinate of the point. So, \(\sin \frac{4\pi}{3} = -\frac{\sqrt{3}}{2}\).
The cosecant function is the reciprocal of sine, so \(\csc t = \frac{1}{\sin t}\). Therefore, \(\csc \frac{4\pi}{3} = \frac{1}{\sin \frac{4\pi}{3}}\).
Substitute the sine value into the reciprocal to express \(\csc \frac{4\pi}{3}\) as \(\csc \frac{4\pi}{3} = \frac{1}{-\frac{\sqrt{3}}{2}}\). Simplify this expression to get the final form.
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Unit Circle and Coordinates
The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Each point on the circle corresponds to an angle t (in radians) measured from the positive x-axis. The coordinates (x, y) of each point represent the cosine and sine of the angle t, respectively.
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Percorso guidato
06:11
Introduction to the Unit Circle
Trigonometric Functions and Their Values
Trigonometric functions such as sine, cosine, and cosecant are defined based on the coordinates of points on the unit circle. For an angle t, sin(t) = y, cos(t) = x, and csc(t) = 1/sin(t). Understanding these relationships allows calculation of function values using the unit circle coordinates.
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6:04Introduction to Trigonometric Functions
Evaluating Cosecant Function
The cosecant function, csc(t), is the reciprocal of sine: csc(t) = 1/sin(t). To find csc(4π/3), first identify the sine value at 4π/3 from the unit circle coordinates, then take its reciprocal. If sin(t) = 0, csc(t) is undefined.
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6:22Graphs of Secant and Cosecant Functions
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.
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.
sec 5𝜋/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>
tan 𝜋/3
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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
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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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