Domanda del libro di testo
In Exercises 9–16, use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.
sin 𝜋/4 - cos 𝜋/4
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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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TOC.tableOfContents
Capitolo 1, Problema 15
The unit circle has been divided into twelve equal arcs, corresponding to t-values of
0, 𝜋/6, 𝜋/3, 𝜋/2, 2𝜋/3, 5𝜋/6, 𝜋, 7𝜋/6, 4𝜋/3, 3𝜋/2, 5𝜋/3, 11𝜋/6, and 2𝜋
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.
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sin 3𝜋/2
Guida verificata passo dopo passo1
Recall that on the unit circle, the coordinates of a point corresponding to an angle \(t\) are given by \((\cos t, \sin t)\).
Identify the angle \(t = 3\pi/2\) on the unit circle. This angle corresponds to rotating \(270^\circ\) counterclockwise from the positive x-axis.
Find the coordinates of the point on the unit circle at \(t = 3\pi/2\). This point lies on the negative y-axis.
Since the sine of an angle is the y-coordinate of the corresponding point on the unit circle, write \(\sin 3\pi/2\) as the y-coordinate of that point.
State the value of \(\sin 3\pi/2\) based on the coordinates found, or note if it is undefined (which it is not in this case).
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Unit Circle and Radian Measure
The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles on the unit circle are measured in radians, where 2π radians correspond to a full rotation of 360°. Dividing the circle into twelve equal arcs means each arc measures π/6 radians, providing standard angle measures for evaluating trigonometric functions.
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Percorso guidato
06:11
Introduction to the Unit Circle
Coordinates on the Unit Circle and Trigonometric Functions
Each point on the unit circle corresponds to an angle t and has coordinates (x, y), where x = cos(t) and y = sin(t). These coordinates allow direct evaluation of sine and cosine values for given angles. For example, sin(3π/2) corresponds to the y-coordinate of the point at angle 3π/2.
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Percorso guidato
6:34Sine, Cosine, & Tangent on the Unit Circle
Definition and Domain of Trigonometric Functions
Sine and cosine functions are defined for all real numbers and correspond to y and x coordinates on the unit circle, respectively. Other functions like tangent are defined as sin(t)/cos(t) and may be undefined where cosine is zero. Understanding when functions are defined or undefined is crucial for correctly evaluating expressions at specific angles.
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Percorso guidato
3:43Finding the Domain of an Equation
Pratica correlata
Domanda del libro di testo
In Exercises 13–17, find a positive angle less than 360° or 2𝜋 that is coterminal with the given angle. -445°
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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.
cos 3𝜋/2
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Domanda del libro di testo
In Exercises 9–16, use the given triangles to evaluate each expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.
tan 𝜋/4 + csc 𝜋/6
635
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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
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.
tan 3𝜋/2
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