Problem 65 - Precalculus: A Concise Course

Question

Convert the angle measure from degrees to radians. Round to three decimal places. $$ 45^{\circ} $$

Step-by-Step Solution

Answer

The conversion of \(45^{\circ}\) into radians will result in approximately \(0.785\) radians when rounded off to three decimal places.

Step 1: Identify the given angle

In this problem, the angle is provided in degrees. The given angle is \(45^{\circ}\).

Step 2: Apply the conversion formula

The formula to convert degrees to radians is \( radians = degrees * \frac{\pi}{180} \). To carry out this conversion, replace 'degrees' with \(45\) in the formula. So the calculation becomes \( radians = 45 * \frac{\pi}{180} \).

Step 3: Simplify the expression

Now, divide 45 by 180, multiply the resulting quotient by \(\pi\), and round off your answer to three decimal places.

Step 4: Compute the answer

The answer after solving the above expression is approximately \(0.785\) radians. Thus, \(45^{\circ} = 0.785\) radians. Round your final answer to three decimal places.

Key Concepts

Angle Measure Conversion

Understanding angle measure conversion is essential when working with different mathematical contexts that require switching between degrees and radians. When you convert an angle from one form to another, you are simply translating the measurement into a different unit. This is akin to converting centimeters to inches or liters to gallons. The basic idea is to maintain the same angle size while expressing it in a different unit.

Degrees are often used in everyday contexts, particularly in education, navigation, and weather forecasts. However, radians are more frequently used in calculus and higher-level mathematics because of their natural connection to the properties of circles and trigonometric functions.

• Degrees: Based on the division of a circle into 360 equal parts.
• Radians: Found by relating the length of a circle's arc to its radius.

Getting comfortable with converting between these two can help bridge the gap between practical applications and theoretical mathematics.

Degree to Radian Formula

The key to converting between degrees and radians lies in the degree to radian formula. This formula is derived from the relationship that \(180\) degrees is equivalent to \(\pi\) radians. Thus, the conversion formula can be expressed as:

\[ radians = degrees \times \frac{\pi}{180} \]

This formula allows for straightforward conversion. You just multiply your angle in degrees by the factor \(\frac{\pi}{180}\).

Example Calculation:
Suppose you want to convert \(45^{\circ}\) to radians. By substituting into the formula, you have:
\[ radians = 45 \times \frac{\pi}{180} \]

Simplifying this expression involves dividing \(45\) by \(180\), which gives \(\frac{1}{4}\), and multiplying by \(\pi\), resulting in \(\frac{\pi}{4}\). This is the exact radian measure. Using the value of \(\pi\) approximately as \(3.14159265359\), you can further solve to obtain a decimal form, rounded to your necessary precision.

Radian Approximation

When you are required to convert degrees to radians, sometimes you need a decimal approximation rather than an exact value. This is often the case when precision is important or when a real-world context requires a readable number.

The process involves estimating \(\pi\) to a recognized value such as \(3.14159265359\). You apply this value in your conversion formula after performing any necessary algebraic simplifications.

• After simplifying your expression, multiply by \(\pi\) and use its value.
• Round the result to as many decimal places as required, ensuring accuracy.

Remember, in our example numbers, converting \(45^{\circ}\) leads to \(\frac{\pi}{4}\), and estimating \(\pi\) to three decimal places will give \(\approx 3.142\). Thus, \(\frac{\pi}{4} \approx 0.785\) radians. Rounding this result correctly is vital to applications needing precision, such as engineering or computer graphics.