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Ch. 3 - Radian Measure and The Unit Circle
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 3 - Radian Measure and The Unit CircleProblema 57
Capitolo 4, Problema 57

Work each problem. See Example 5. Angle Measure Find the measure (in radians) of a central angle of a sector of area 16 in² a circle of radius 3.0 in.

Guida verificata passo dopo passo
1
Recall the formula for the area of a sector of a circle: \(A = \frac{1}{2} r^{2} \theta\), where \(A\) is the area of the sector, \(r\) is the radius, and \(\theta\) is the central angle in radians.
Identify the given values: the area \(A = 16\) in² and the radius \(r = 3.0\) in.
Substitute the known values into the formula: \(16 = \frac{1}{2} \times (3.0)^{2} \times \theta\).
Simplify the expression on the right side: calculate \(\frac{1}{2} \times 9 = 4.5\), so the equation becomes \(16 = 4.5 \times \theta\).
Solve for \(\theta\) by dividing both sides of the equation by 4.5: \(\theta = \frac{16}{4.5}\).

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Area of a Sector

The area of a sector of a circle is given by the formula A = (1/2) * r² * θ, where r is the radius and θ is the central angle in radians. This formula relates the sector's area directly to the angle, allowing calculation of one when the other is known.
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Calculating Area of SAS Triangles

Radian Measure of Angles

Radians measure angles based on the radius of a circle, where one radian is the angle subtended by an arc equal in length to the radius. Using radians simplifies many trigonometric formulas, especially those involving arc length and sector area.
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Converting between Degrees & Radians

Solving for the Central Angle

To find the central angle θ when the sector area and radius are known, rearrange the sector area formula to θ = (2 * A) / r². This step involves algebraic manipulation and understanding the relationship between the variables.
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2:28
Solving Problems with Complementary & Supplementary Angles
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