Problem 38 - Algebra and Trigonometry

Question

In Exercises \(35-40,\) convert each angle in radians to degrees. Round to two decimal places. \(\frac{\pi}{17}\) radians

Step-by-Step Solution

Answer

The angle in degrees equivalent to \(\frac{\pi}{17}\) radians is approximately \(10.59\)\(^\circ\).

Step 1: Understand the conversion formula

Firstly, one must convert radians to degrees. This is done using the formula \(degrees = radians * \frac{180}{\pi}\)

Step 2: Substitute the given radians into the formula

After that, substitute the given radian measurement \(\frac{\pi}{17}\) into the formula: \(degrees = \frac{\pi}{17} * \frac{180}{\pi}\)

Step 3: Simplify the equation

Simplify the equation by cancelling out the common terms in the numerator and denominator. Here, \(\pi\) in the numerator cancels out \(\pi\) in the denominator giving \(degrees = \frac{180}{17}\)

Step 4: Calculate the final result

Finally, calculate the result by dividing \(180\) by \(17\). Round it off to two decimal places considering the instruction in the exercise.

Key Concepts

Angle Measurement

Angle measurement can be a bit tricky. There are two primary ways angles are measured: in degrees and in radians. Degrees are more familiar to most, as they divide a circle into 360 equal parts. In contrast, radians measure angles using the radius of a circle.
A full circle encompasses \(2\pi\) radians. This means half a circle (or 180°) is \(\pi\) radians.
Therefore, understanding radians is crucial as they are used extensively in mathematics, especially in calculus and trigonometry.

• Radians offer a way to relate angles directly to the properties of a circle.
• Degrees are often more intuitive for everyday use.

Recognizing the difference and knowing how to convert between these measurements is essential.

Conversion Formula

When you need to convert radians to degrees, there's a handy conversion formula you can use. This formula is based on the equivalence of \(\pi\) radians being equal to 180 degrees.
The conversion formula is:\[degrees = radians \times \frac{180}{\pi}\]

• It allows you to translate any radian measure into degrees.
• You multiply the radian value by \(\frac{180}{\pi}\) to find the equivalent degree measure.

This formula comes in handy when working with angles in various contexts, enabling easy switches between different units of measurement.

Simplifying Equations

Simplifying an equation is an important step in any mathematical problem-solving process. In the conversion formula for radians to degrees, simplifying the equation helps make calculations easier and eliminates unnecessary steps.
Consider the example given in the exercise: \[degrees = \frac{\pi}{17} \times \frac{180}{\pi}\]
By canceling out the \(\pi\) in the numerator and denominator, you simplify to \[degrees = \frac{180}{17}\].
This simplification reduces the complexity of the calculation.

• Identifying common terms that can be canceled makes the math more straightforward.
• Simplification helps focus on the core numbers you need to calculate the final result.

Rounding Decimals

Rounding decimals is often the last step in calculations, especially when asked for a specific number of decimal places. It’s a crucial skill because it provides a tidy, usable number that's easier to interpret.
For the final step in this equation, \(\frac{180}{17}\) yields approximately 10.58823. Rounding this to two decimal places gives you \(10.59\).
This is done by looking at the third decimal place; if it's 5 or higher, round the second decimal up, otherwise round down.

• Rounding creates practical numbers that are simpler to read.
• Always check the instructions for how many decimal places are needed.

Mastering rounding keeps numbers clear and precise in your results.