Problem 70 - Precalculus: A Concise Course

Question

Convert the angle measure from degrees to radians. Round to three decimal places. $$ 345^{\circ} $$

Step-by-Step Solution

Answer

The angle \( 345^\circ \) is equivalent to approximately \( 6.022 \) radians when rounded to three decimal places.

Step 1: Identify the given angle in degrees

The given angle is \( 345^\circ \).

Step 2: Apply the conversion formula

We apply the formula \( \frac{π}{180} \times 345 \) to convert degrees to radians.

Step 3: Calculate and round the result

The result is approximately 6.021, but we need to round it to 3 decimal places. Therefore, the answer is \( 6.022 \) radians.

Key Concepts

Angle Conversion: Understanding the Basics

Angle conversion is a crucial part of understanding and working with circles, trigonometric functions, and geometry concepts. By converting angles from degrees to radians, we can better utilize mathematical functions and equations that require angles in radians.

Degrees and radians are two different units of measuring angles. While degrees are more commonly used in everyday situations, radians are preferred in higher mathematics and sciences. Remember:

• Degrees are represented as parts of a full circle, which equals 360 degrees.
• Radians, however, use the circle's radius and are based on the mathematical constant \( \pi \).

To convert an angle from degrees to radians, you use the conversion formula: \[ \text{Radians} = \frac{\pi}{180} \times \text{Degrees} \] The formula essentially scales the degree measure using \( \pi \), allowing it to convert into radians. For example, converting \(345^\circ\) involves plugging into the formula to get approximately 6.021 radians, which we further round to 6.022 radians for precision.

Trigonometric Functions: Link to Angle Measurement

Trigonometric functions rely heavily on angle measurements and are essential in calculations involving right triangles, waves, and oscillations. These functions often use radians in their calculations, reinforcing the need for understanding degree to radian conversion.

Some key trigonometric functions include:

• \(\sin(\theta)\) - Sine of the angle.
• \(\cos(\theta)\) - Cosine of the angle.
• \(\tan(\theta)\) - Tangent of the angle.

Radians are used in trigonometry for a few important reasons:

First, they simplify the derivatives of these functions in calculus.

Second, since radians directly relate to the circle's radius, they provide a more natural, mathematical means to describe periodic functions. This factor is important in fields such as physics and engineering.

Mathematical Constants: The Importance of π

The mathematical constant \( \pi \) plays a significant role in geometry and trigonometry, especially when converting angles.

\( \pi \) represents the ratio of a circle's circumference to its diameter, a constant that appears in many formulas involving circles. Its approximate value is 3.14159, but it's represented as \( \pi \) because it is an irrational number and cannot be precisely expressed as a decimal.

In angle conversions:

\( \pi \) is essential because it standardizes the relationship between the radius of a circle and its different arcs, essentially converting a degree measure into a portion of the circle's circumference.

Using \( \pi \) also results in elegantly simple expressions for trigonometric properties and limits the hassle of cumbersome calculations. In fact, the conversion formula \( \frac{\pi}{180} \) x Degrees demonstrates this importance clearly, converting degrees into radians efficiently.