Domanda del libro di testo
In Exercises 8–12, draw each angle in standard position. 5𝜋 6
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Ch. 1 - Angles and the Trigonometric Functions
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Capitolo 1, Problema 7
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: 10 inches Arc Length, s: 40 inches
Guida verificata passo dopo passo1
Recall the formula that relates the arc length \(s\), the radius \(r\), and the central angle \(\theta\) in radians:
\[ s = r \times \theta \]
Identify the given values from the problem: radius \(r = 10\) inches and arc length \(s = 40\) inches.
Substitute the known values into the formula:
\[ 40 = 10 \times \theta \]
Solve for the central angle \(\theta\) by dividing both sides of the equation by the radius \(r\):
\[ \theta = \frac{40}{10} \]
Simplify the fraction to express \(\theta\) in radians, which will give the radian measure of the central angle.
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Radian Measure of an Angle
A radian is a unit of angular measure based on the radius of a circle. One radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius. It provides a natural way to relate angles to arc lengths without converting to degrees.
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Percorso guidato
5:04
Converting between Degrees & Radians
Relationship Between Arc Length, Radius, and Central Angle
The central angle θ (in radians) of a circle is related to the arc length s and radius r by the formula θ = s / r. This formula allows you to find the angle when the arc length and radius are known, making it essential for solving problems involving circular arcs.
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04:33Find the Angle Between Vectors
Units and Conversion in Trigonometry
Understanding units is crucial; arc length is measured in linear units (e.g., inches), radius in the same units, and the central angle in radians (a dimensionless measure). Ensuring consistent units and interpreting the result in radians is key to correctly solving and applying trigonometric problems.
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6:34Sine, Cosine, & Tangent on the Unit Circle
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In Exercises 5–7, convert each angle in radians to degrees. - 5𝜋 6
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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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cos 5𝜋/6
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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. sin 𝜋/3
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In Exercises 5–7, convert each angle in radians to degrees. 7𝜋 5
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In Exercises 1–6, the measure of an angle is given. Classify the angle as acute, right, obtuse, or straight. 𝜋/2
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