Problem 34 - Precalculus

Question

Convert from radians to degrees. $$\frac{11 \pi}{9}$$

Step-by-Step Solution

Answer

The angle in degrees is 220°.

Step 1: Understand the Relationship

To convert from radians to degrees, use the relation: \( 180^{\circ} = \pi \text{ radians} \). This means that 1 radian is equal to \( \frac{180}{\pi} \text{ degrees} \).

Step 2: Set Up the Conversion

Use the relationship from Step 1 to convert \( \frac{11 \pi}{9} \) radians to degrees by multiplying: \( \frac{11 \pi}{9} \times \frac{180}{\pi} \).

Step 3: Cancel Out \(\pi\)

When multiplying, the \(\pi\) in the numerator and denominator will cancel each other out, simplifying the expression to: \( \frac{11 \times 180}{9} \).

Step 4: Perform the Multiplication

Calculate \(11 \times 180 = 1980\).

Step 5: Simplify the Expression

Divide 1980 by 9 to get the answer in degrees: \( \frac{1980}{9} = 220 \text{ degrees} \).

Key Concepts

Radian

Radians are a way of measuring angles based on the radius of a circle. They offer a more natural approach to angle measurement, especially in mathematics and physics, where many formulas involve trigonometric functions.
In this system, the angle is defined by the length of the arc of a circle that subtends the angle at its center. One radian is the angle created when the radius is wrapped along the edge of the circle.

• One complete revolution around a circle is equal to 2π radians.
• A straight angle, which is half a rotation, measures π radians.
• A right angle, being a quarter of a rotation, accounts for π/2 radians.

Using radians simplifies many mathematical equations, as these naturally incorporate the properties of the circle itself.

Degree

Degrees are perhaps the most familiar way to measure angles, especially in practical applications like navigation and construction. This system divides a full circle into 360 equal parts.
A big advantage of degrees is their simplicity when dealing with right angles and straight lines. For instance:

• A full circle equals 360 degrees.
• A straight line measures 180 degrees.
• A right angle, commonly encountered in many contexts, is 90 degrees.

The division into 360 parts has historical roots, stemming from ancient civilizations that used this number for its numerous divisors, making calculations easier in everyday scenarios. Degrees offer an intuitive feel for common angles found in polygons and work well with systems based on base-10 mathematics.

Angle Measurement

Angle measurement is crucial for understanding geometric relationships and trigonometric calculations. It is about gauging how much one line or plane is tilted relative to another. Being able to express angles in different ways allows for flexibility when dealing with mathematical problems.
There are two main units for angle measurement: radians and degrees.

• Radians are preferred in calculus and higher mathematics due to their natural relationship with circular functions.
• Degrees are more straightforward in everyday applications and basic geometry.

To convert between these two, one needs to understand that 180 degrees equals π radians. This relationship enables converting measurements from one unit to the other, for instance, using multiplication and division in simple calculations, just like in the exercise you might encounter.