Problem 8 - Algebra 2

Question

Rewrite each degree measure in radians and each radian measure in degrees. \(\frac{3 \pi}{4}\)

Step-by-Step Solution

Answer

\( \frac{3 \pi}{4} \) radians is 135 degrees.

Step 1: Understanding the Conversion

To convert radians to degrees, we use the formula: degrees = radians × \( \frac{180}{\pi} \). This formula is derived from the relationship that \( \pi \, \text{radians} = 180^{\circ} \).

Step 2: Applying the Formula

Given \( \frac{3 \pi}{4} \) radians, substitute this into the formula: \( \text{degrees} = \frac{3 \pi}{4} \times \frac{180}{\pi} \).

Step 3: Canceling Out \( \pi \) and Simplifying

Cancel \( \pi \) in the numerator and denominator: \( \frac{3}{4} \times 180 \). This simplifies to \( \frac{540}{4} \).

Step 4: Calculating the Final Result

Divide 540 by 4 to get 135. So, \( \frac{3 \pi}{4} \) radians is equal to 135 degrees.

Key Concepts

Radian to Degree Conversion

Converting between radians and degrees is a fundamental skill in trigonometry. Understanding this concept involves a simple mathematical relationship where π radians is equivalent to 180 degrees. To convert radian measures into degrees, you can use the formula: degrees = radians × \( \frac{180}{\pi} \).
Thus, the conversion process involves multiplying the radian measure by \( \frac{180}{\pi} \).
For students, keeping this conversion factor in mind simplifies many problems and helps move seamlessly between these two units of measurement.

Degree Measure

Degrees are a common unit for measuring angles, especially in everyday use. A full circle consists of 360 degrees, making it intuitive and straightforward to understand.
When dealing with angles in geometry and various practical applications, using degrees can often make calculations easier. Also, degrees are used extensively in disciplines such as navigation and engineering.
Understanding the degree measure allows one to easily shift into more complex trigonometric concepts as required.

Radian Measure

Radians offer an alternative way to measure angles and are particularly important in higher mathematics and many physics applications.
Instead of dividing a circle into 360 equal parts, radians represent the angle as the radius arcs in a unit circle. Hence, a full circle is \( 2\pi \) radians.
This method is rooted in the properties of the circle itself, making it quite natural for mathematical conversions and analyses. Grasping radians helps in understanding periodic functions and calculus concepts.

Trigonometry

Trigonometry is the mathematics of angles and has real-world applications in fields like science, engineering, and architecture.
Understanding trigonometric functions requires familiarity with both radians and degrees. The functions sine, cosine, and tangent, among others, intertwine these two units in their calculations.
Proficiency in trigonometry facilitates solving triangles and comprehending wave patterns, lending to its multifaceted applications in the real world.

Mathematical Conversion

Mathematical conversion is the process of changing from one unit to another and is essential in numerous areas of math and science.
For angle conversion, this usually involves either converting radians to degrees or vice-versa using the conversion formula where \( \pi \) radians equals 180 degrees.
Mastery of these conversions aids in understanding and working with various mathematical principles, ensuring competence in areas like calculus and geometry.