Domanda del libro di testo
In Exercises 17–20, θ is an acute angle and sin θ and cos θ are given. Use identities to find tan θ, csc θ, sec θ, and cot θ. Where necessary, rationalize denominators.
sin θ = 3/5, cos θ = 4/5
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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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TOC.tableOfContents
Capitolo 1, Problema 20
In Exercises 19–24,
a. Use the unit circle shown for Exercises 5–18 to find the value of the trigonometric function.
b. Use even and odd properties of trigonometric functions and your answer from part (a) to find the value of the same trigonometric function at the indicated real number.
cos 𝜋/3
Guida verificata passo dopo passo1
Locate the angle \( \frac{\pi}{3} \) on the unit circle. It corresponds to the point \( \left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right) \).
Identify the cosine value of \( \frac{\pi}{3} \) from the unit circle, which is the x-coordinate of the point, \( \frac{1}{2} \).
Recall the even property of cosine: \( \cos(-x) = \cos(x) \).
Use the even property to find \( \cos(-\frac{\pi}{3}) \), which is also \( \frac{1}{2} \).
Conclude that the cosine value for both \( \frac{\pi}{3} \) and \( -\frac{\pi}{3} \) is \( \frac{1}{2} \).
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Unit Circle
The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is fundamental in trigonometry as it provides a geometric representation of the sine, cosine, and tangent functions. The coordinates of points on the unit circle correspond to the cosine and sine values of angles measured from the positive x-axis, allowing for easy calculation of trigonometric functions.
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Percorso guidato
06:11
Introduction to the Unit Circle
Trigonometric Functions
Trigonometric functions, such as sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. In the context of the unit circle, the cosine of an angle is the x-coordinate, while the sine is the y-coordinate of the corresponding point on the circle. Understanding these functions is crucial for solving problems involving angles and their relationships in various mathematical contexts.
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Percorso guidato
6:04Introduction to Trigonometric Functions
Even and Odd Properties
Trigonometric functions exhibit specific symmetry properties: cosine is an even function, meaning cos(-x) = cos(x), while sine and tangent are odd functions, meaning sin(-x) = -sin(x) and tan(-x) = -tan(x). These properties allow for simplifications when calculating trigonometric values for negative angles or angles greater than π, making it easier to find values using known angles on the unit circle.
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Percorso guidato
06:19Even and Odd Identities
Pratica correlata
Domanda del libro di testo
In Exercises 19–24,
a. Use the unit circle shown for Exercises 5–18 to find the value of the trigonometric function.
b. Use even and odd properties of trigonometric functions and your answer from part (a) to find the value of the same trigonometric function at the indicated real number.
cos (-𝜋/6)
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Domanda del libro di testo
In Exercises 21–24, θ is an acute angle and sin θ is given. Use the Pythagorean identity sin²θ + cos²θ = 1 to find cos θ.
sin θ = 6/7
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Domanda del libro di testo
In Exercises 17–20, θ is an acute angle and sin θ and cos θ are given. Use identities to find tan θ, csc θ, sec θ, and cot θ. Where necessary, rationalize denominators.
__
sin θ = 6, cos θ = √13
7 7
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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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Domanda del libro di testo
In Exercises 19–24,
a. Use the unit circle shown for Exercises 5–18 to find the value of the trigonometric function.
b. Use even and odd properties of trigonometric functions and your answer from part (a) to find the value of the same trigonometric function at the indicated real number.
sin 5𝜋/6
1361
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