Problem 120 - Calculus Volume 1

Question

Convert each angle in radians to degrees. \(\frac{11 \pi}{2} \mathrm{rad}\)

Step-by-Step Solution

Answer

990 degrees

Step 1: Understand the Conversion Formula

To convert from radians to degrees, we use the formula: \[D = R \times \left( \frac{180}{\pi} \right)\]where \(D\) is the angle in degrees, and \(R\) is the angle in radians.

Step 2: Substitute the Given Value

Substitute the given value \( R = \frac{11\pi}{2} \) into the conversion formula:\[D = \frac{11\pi}{2} \times \left( \frac{180}{\pi} \right)\]

Step 3: Simplify the Expression

The \(\pi\) terms cancel out in the expression, and we simplify further:\[D = \frac{11}{2} \times 180\]

Step 4: Perform the Multiplication

Calculate the product of \(\frac{11}{2}\) and 180:\[D = 11 \times 90 = 990\]

Step 5: State the Final Result

The angle \(\frac{11 \pi}{2}\) radians is equivalent to 990 degrees.

Key Concepts

Angle Conversion

Converting angles between radians and degrees is a fundamental skill in trigonometry and mathematics. Understanding how these two measurement systems relate helps simplify many problems.
A radian is another way to measure angles. It is based on the radius of a circle. In simple terms, one radian is the angle created when the arc length is equal to the circle's radius. This system is particularly convenient in calculus and higher-level mathematics.
To convert an angle from radians to degrees, we use the conversion factor \(\frac{180}{\pi}\) .

• The number \(\pi\) is approximately 3.14159 and represents the ratio of the circumference of any circle to its diameter.
• Multiplying radians by \(\frac{180}{\pi}\) transforms the angle unit from radians to degrees.

This conversion is vital because degrees are more intuitive. Most people understand degrees in the context of geometry. Remember, a full circle is 360 degrees or \(2\pi\) radians.

Trigonometry

Trigonometry deals with the study of angles and the relationships between them. It originates from the geometry of triangles, especially right-angled ones.
Understanding trigonometry is essential for many applications in science and engineering. It extends to circular functions, which involve angles and lengths on the radius of a circle.
The use of radians in trigonometry is widespread as they simplify calculations. Consider that many of the trigonometric identities and formulas become more concise and simpler when expressed with radians.

• Sine, cosine, and tangent functions (and their inverses) are better defined through radians.
• Radians align better with periodic functions like sine and cosine, which have natural periods of \(2\pi\) .

In essence, radians provide a natural way to work with angles in many mathematical contexts, enhancing the precision and simplicity of trigonometric calculations.

Mathematical Calculations

Mathematical calculations involving angle conversion require careful attention to units and precision. Here's how we perform such a calculation in detail.
First, identify the angle given in radians and note the conversion formula \(D = R \times \left( \frac{180}{\pi} \right)\) .

• Substitute the given radian value into the expression. For example, if \(R = \frac{11\pi}{2} \), inserting this value gives \(D = \frac{11\pi}{2} \times \left( \frac{180}{\pi} \right)\) .
• Cancel the \(\pi\) from the expression, reducing it to \(D = \frac{11}{2} \times 180\) .
• Perform the multiplication: \(D = 11 \times 90\) , which calculates to 990.
• The result means the angle \(\frac{11\pi}{2}\) radians converts to 990 degrees.

Through each step, attention to detail ensures accuracy. This method can be applied to other radian values for efficient problem-solving.