In Exercises 71–74, find the length of the arc on a circle of radius r intercepted by a central angle θ. Express arc length in terms of 𝜋. Then round your answer to two decimal places.Radius, r: 12 inchesCentral Angle, θ: θ = 45°

Verified step by step guidance

1. Convert the central angle from degrees to radians using the formula: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \).
2. Substitute \( \theta = 45^\circ \) into the formula to find the angle in radians.
3. Use the formula for arc length: \( s = r \times \theta \), where \( s \) is the arc length, \( r \) is the radius, and \( \theta \) is the angle in radians.
4. Substitute \( r = 12 \) inches and the calculated \( \theta \) in radians into the arc length formula.
5. Simplify the expression to find the arc length in terms of \( \pi \), and then round the result to two decimal places.

Key Concepts

Arc Length Formula

The arc length of a circle can be calculated using the formula L = rθ, where L is the arc length, r is the radius, and θ is the central angle in radians. To use this formula, it is essential to convert the angle from degrees to radians, as the standard unit for angles in trigonometry is radians.

Conversion from Degrees to Radians

To convert an angle from degrees to radians, you can use the conversion factor π radians = 180 degrees. Therefore, to convert degrees to radians, multiply the degree measure by π/180. For example, a 45° angle converts to (45 × π/180) radians, which simplifies to π/4 radians.

Expressing in Terms of π

When asked to express the arc length in terms of π, it means to leave the answer in a form that includes π rather than calculating a decimal approximation. This is often useful in mathematics to maintain precision and clarity, especially when dealing with circular measurements.