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Ch. 1 - Angles and the Trigonometric Functions
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 3rd EditionCh. 1 - Angles and the Trigonometric FunctionsProblema 9
Capitolo 1, Problema 9

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: 6 yards Arc Length, s: 8 yards

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Recall the formula that relates the arc length \( s \), radius \( r \), and central angle \( \theta \) in radians: \[ s = r \times \theta \].
Identify the given values: radius \( r = 6 \) yards and arc length \( s = 8 \) yards.
Substitute the known values into the formula: \[ 8 = 6 \times \theta \].
Solve for the central angle \( \theta \) by dividing both sides of the equation by 6: \[ \theta = \frac{8}{6} \].
Simplify the fraction if possible to express \( \theta \) in radians.

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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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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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Units Consistency in Trigonometric Calculations

When calculating angles and lengths in trigonometry, it is important to ensure that all measurements are in consistent units. Here, both radius and arc length are given in yards, so no unit conversion is needed, simplifying the calculation of the radian measure.
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Sine, Cosine, & Tangent on the Unit Circle
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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.

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