Problem 21 - Algebra and Trigonometry

Question

In Exercises \(21-28,\) convert each angle in radians to degrees. $$ \frac{\pi}{2} $$

Step-by-Step Solution

Answer

The angle of \(\frac{\pi}{2}\) radians in degrees is \(90°\).

Step 1: Identify the given

First of all it's important to recognize that the angle is given as \(\frac{\pi}{2}\) radians

Step 2: Use the Conversion Formula

We know that \(\pi\) radians is equal to 180 degrees. This information allows us to build a conversion factor. Using this, the conversion can be done as follows: \( \frac{\pi}{2} \text{ rad} \times \frac{180°}{\pi \text{ rad}} \)

Step 3: Calculate the Result

Now, just need to do the multiplication: \( \frac{\pi}{2} \times \frac{180}{\pi} = \frac{180}{2} = 90° \)

Key Concepts

Radians to Degrees

Radians and degrees are two different units of measuring angles. The radian is a unit based on the radius of a circle, while a degree is \(1/360\) of a full circle. Understanding how to convert between these units is essential in various fields of mathematics, especially in trigonometry and calculus. When converting from radians to degrees, we need to apply a specific conversion factor, because these units are not directly comparable. This conversion is particularly important for solving geometry problems involving circles since angles are often measured in either degrees or radians depending on the context.

Conversion Factor

To convert an angle measurement from radians to degrees, we rely on a crucial conversion factor. This factor is based on the relationship that \(\pi\) radians are equivalent to 180 degrees.

• This means that one radian is approximately \(57.3\) degrees (calculated by taking \(180\/\pi\)).
• The conversion factor from radians to degrees is given by the formula: \[ 180^\circ\/\pi. \] This ratio helps us transition seamlessly between these two units.

To perform the conversion, multiply the angle in radians by \(180^\circ\/\pi\), as shown in the original exercise. This converts the measurement into degrees effectively.

Mathematical Calculations

The process of mathematical calculations involved in angle conversion is straightforward but requires attention to detail. After identifying the given angle in radians and setting up the conversion using the conversion factor, the next step is to carry out the multiplication. For example, when converting \(\frac{\pi}{2}\) radians to degrees:Start by setting up the multiplication: \(rac{\pi}{2} \times \frac{180}{\pi}\). Cancel out the \(\pi\) in the numerator and denominator:

• This results in \(\frac{180}{2}\).

Finally, perform the division to get the angle in degrees: \(\frac{180}{2} = 90^\circ\). This illustrates how the formula offers a direct yet precise method for conversion.

Trigonometry Concepts

Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. Many trigonometric problems use angles measured in both degrees and radians, so understanding how to convert between these units is vital. In trigonometry:

• Radian measures are frequently used because they can simplify many mathematical expressions, especially in calculus.
• Degrees are often more intuitive for practical applications, like navigation and engineering.

When solving trigonometric problems, converting angles from radians to degrees and vice versa can make calculations easier and help in visualizing the problem. Thus, mastering these conversions enhances your problem-solving toolkit, providing flexibility in handling various mathematical challenges.