Domanda del libro di testo
In Exercises 49–59, find the exact value of each expression. Do not use a calculator. tan 120°
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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Blitzer 3rd EditionCh. 1 - Angles and the Trigonometric FunctionsProblema 1.22a
Blitzer 3rd EditionCh. 1 - Angles and the Trigonometric FunctionsProblema 1.22aCapitolo 1, Problema 1.22a
Use the unit circle shown to find the value of the trigonometric function.
sin (2𝜋/3)
Guida verificata passo dopo passo1
Step 1: Recognize that the angle \( \frac{2\pi}{3} \) is in radians and corresponds to 120 degrees.
Step 2: Identify the position of \( \frac{2\pi}{3} \) on the unit circle. It is in the second quadrant.
Step 3: Recall that in the second quadrant, the sine function is positive.
Step 4: Use the reference angle for \( \frac{2\pi}{3} \), which is \( \pi - \frac{2\pi}{3} = \frac{\pi}{3} \).
Step 5: The sine of \( \frac{2\pi}{3} \) is the same as the sine of its reference angle \( \frac{\pi}{3} \), which is \( \sin(\frac{\pi}{3}) \).
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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 a fundamental tool in trigonometry, as it allows for the definition of trigonometric functions based on angles measured from the positive x-axis. Each point on the unit circle corresponds to a specific angle and provides the sine and cosine values for that angle, which are essential for evaluating trigonometric functions.
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Percorso guidato
06:11
Introduction to the Unit Circle
Sine Function
The sine function, denoted as sin(θ), is a trigonometric function that represents the y-coordinate of a point on the unit circle corresponding to an angle θ. For angles in the second quadrant, such as 2π/3, the sine value is positive. Understanding the sine function's behavior in different quadrants is crucial for accurately determining its value for various angles.
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5:53Graph of Sine and Cosine Function
Angle Measurement in Radians
In trigonometry, angles can be measured in degrees or radians, with radians being the standard unit in mathematical contexts. The angle 2π/3 radians corresponds to 120 degrees, placing it in the second quadrant of the unit circle. Recognizing how to convert between degrees and radians and understanding the implications of angle placement on the unit circle is vital for evaluating trigonometric functions.
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5:04Converting between Degrees & Radians
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Domanda del libro di testo
Use the unit circle shown to find the value of the trigonometric function.
cos 𝜋/6
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Domanda del libro di testo
Use the unit circle shown to find the value of the trigonometric function.
tan 11𝜋/6
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Domanda del libro di testo
In Exercises 1–4, a point P(x, y) is shown on the unit circle corresponding to a real number t. Find the values of the trigonometric functions at t.
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Domanda del libro di testo
The unit circle has been divided into eight equal arcs, corresponding to t-values of
0, 𝜋/4, 𝜋/2, 3𝜋/4, 𝜋, 5𝜋/4, 3𝜋/2, 7𝜋/4, and 2𝜋.
a. Use the (x,y) coordinates in the figure to find the value of the trigonometric function.
b. Use periodic properties and your answer from part (a) to find the value of the same trigonometric function at the indicated real number.
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sin 3𝜋/4
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Domanda del libro di testo
In Exercises 2–4, convert each angle in degrees to radians. Express your answer as a multiple of 𝜋. 15°
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