Problem 115 - Calculus Volume 1

Question

Convert each angle in degrees to radians. Write the answer as a multiple of \(\pi\). $$-60^{\circ}$$

Step-by-Step Solution

Answer

The angle \(-60^{\circ}\) converts to \(-\frac{\pi}{3}\) radians.

Step 1: Understand the Conversion Formula

To convert an angle from degrees to radians, use the formula: \( ext{radians} = ext{degrees} \times \frac{\pi}{180} \). This formula derives from the fact that \(180^{\circ}\) is equivalent to \( \pi \) radians.

Step 2: Substitute the Given Degree Value

Substitute \(-60\) for degrees in the conversion formula: \( -60 \times \frac{\pi}{180} \). This sets up the equation to solve for the radian measurement.

Step 3: Perform the Multiplication

Perform the multiplication \(-60 \times \frac{\pi}{180} = \frac{-60\pi}{180}\). This simplifies the expression to include \(\pi\) as a factor.

Step 4: Simplify the Fraction

Simplify the fraction \(\frac{-60\pi}{180}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 60. This results in the simplified fraction \(\frac{-\pi}{3}\).

Step 5: Write the Final Answer

The angle \(-60^{\circ}\) is equivalent to \(-\frac{\pi}{3}\) radians. This is your final answer expressed as a multiple of \(\pi\).

Key Concepts

Angle Conversion

Converting angles from degrees to radians is an essential skill in mathematics, especially in trigonometry and calculus. This conversion is necessary because many mathematical computations involving curves and rotation are expressed naturally in radians. The conversion process is straightforward and follows a specific formula.

• Degrees and radians are two units for measuring angles, similar to how inches and centimeters measure length.
• One full circle is 360 degrees but only \(2\pi\) radians, making radians a more natural measurement in math.
• The conversion formula is: \(\text{radians} = \text{degrees} \times \frac{\pi}{180}\).

Breaking down the formula, \(180^{\circ}\) is equivalent to \(\pi\) radians, a relationship derived from the fact that the circumference of a circle is \(2\pi\times\text{radius}\). Hence, multiplying degrees by \(\frac{\pi}{180}\) converts the degree measure into radians, effectively rescaling it for the radian system.

Radian Measurement

Radian is a unit of angle that provides a direct method to relate angles with circles and their arcs. The radian allows mathematicians and engineers to express angles through a different perspective compared to degrees.

• One radian is the angle created when the radius of a circle is wrapped along its circumference.
• This makes radian measurement intrinsic to circular motion, as it relates directly to arc length.
• Because \(\pi\) radians equal 180 degrees, using radians keeps calculations simpler and more direct.

Radian measurement streamlines mathematical equations, like those found in physics, as it incorporates \(\pi\), the fundamental constant representing this unique circle relationship.Radians are often used in calculus to represent periodic functions such as sine and cosine, due to the simplicity they bring to differentiation and integration.

Mathematical Formula

Formulas play a crucial role in mathematical problem solving, providing steps to achieve conversions like degrees to radians. Understanding how to dissect and apply these formulas is important for mastering different math concepts.

• To convert degrees to radians, the formula \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \) encapsulates the entire process.
• By substituting the given degree value, you can easily set up the equation.
• Consider simplifying fractions for neat and accurate results since radians are often expressed in simple terms of \(\pi\).

Let's apply it: if you have an angle like \(-60^\circ\), substituting \(-60\) into the formula gives \( -60 \times \frac{\pi}{180} \), which simplifies to \( -\frac{\pi}{3} \). This illustrates how leveraging mathematical formulas can make calculations like angle conversion more approachable and systematic.