Problem 72 - Algebra 2

Question

Write each measure in radians. Express the answer in terms of \(\pi\) and as a decimal rounded to the nearest hundredth. $$ 70^{\circ} $$

Step-by-Step Solution

Answer

The radian measure of \(70^{\circ}\) is \( \frac{7\pi}{18}\) radians or approximately 1.22 radian.

Step 1: Understand the Conversion Factor

Given the conversion factor between degrees and radians which is \( \frac{\pi}{180}\) radians per degree, it provides the foundational understanding that each degree is equivalent to \( \frac{\pi}{180}\) radians. This understanding will be used in subsequent steps.

Step 2: Apply the Conversion Factor

Now, we apply the conversion factor to the given measure in degrees. We multiply 70 degrees by the conversion factor \( \frac{\pi}{180}\), and this will give us the measure in radians.

Step 3: Simplify the Result

After multiplication, simplify the result if needed. This simplification will lead to the desired solution.

Step 4: Decimal Representation

Furthermore, we can obtain a decimal representation of the result by substituting \( \pi\) for its numerical approximate which is approximately 3.1416, then round it to the nearest hundredth.

Key Concepts

trigonometry

Trigonometry is a branch of mathematics that explores the relationships between the angles and sides of triangles. These relationships are pivotal in various applications, from engineering to physics. The core of trigonometry lies in understanding the measures of angles and how they interact with the lengths of triangle sides.

• Angles can be oriented in different ways, most commonly in degrees or radians.
• Degrees are a more familiar unit, where a full circle is 360 degrees.
• Radians provide a more natural measure in mathematical analysis.

In trigonometry, converting degrees to radians is fundamental, as radians are often used in calculus and more advanced mathematics because they simplify numerous mathematical formulas.

pi as a constant

Pi, denoted as \( \pi \), is an irrational number approximately equal to 3.14159. It represents the ratio of a circle's circumference to its diameter. This constant is indispensable in trigonometry and geometry.

In converting degrees to radians, \( \pi \) is integral to the process.

One full circle is \( 2\pi \) radians, demonstrating the relationship between \( \pi \) and circular measurements.

By using \( \pi \), calculations involving angles become more precise.

Utilizing \( \pi \) not only helps in conversions but also allows us to express angles in a form that aligns with many mathematical fields, enhancing accuracy and consistency.

angle measurement

Understanding angle measurement is crucial in applying trigonometry. Angles can be measured in degrees or radians, and each has its specific uses.

• Degrees are more intuitive; we often learn them first.
• Radians are invented precisely for mathematical simplicity, especially in calculus.
• One degree equals \( \frac{\pi}{180} \) radians.

This conversion factor is key. When given an angle like \( 70^{\circ} \), using the conversion formula \( \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \), you can convert it to radians (\( \approx 1.22 \) radians when expressed as a decimal). This process simplifies many complex mathematical problems by transitioning to a consistent unit of measure, making advanced computations more manageable.