Problem 20 - Algebra and Trigonometry

Question

In Exercises \(13-20,\) convert each angle in degrees to radians. Express your answer as a multiple of \(\pi\). $$ -270^{\circ} $$

Step-by-Step Solution

Answer

The angle -270 degrees in radians is \(-\frac{3}{2}\pi \)

Step 1: Write down the conversion formula

The formula to convert degrees to radians is given by: \(Radians = Degrees \times \frac{\pi}{180}\)

Step 2: Substitute the given degree into the formula

Substitute -270 degrees into the formula: Radians = -270 \( \times \frac{\pi}{180}\)

Step 3: Simplify the equation

On simplifying, the answer is given by: \(-\frac{3}{2}\pi \)

Key Concepts

Radian Measure

The radian measure is a way of expressing angles that is particularly useful in mathematics, especially in trigonometry and calculus. Unlike degrees, which divide a circle into 360 arbitrary units, radians are based on the radius of the circle. One radian is the angle created when we wrap the radius of a circle around its circumference.

Since the circumference of a circle is \(2\pi r\), where \(r\) is the radius, there are \(2\pi\) radians in a full circle. This means \(360^{\text{o}}\) is equivalent to \(2\pi\) radians. This fundamental relationship allows us to convert between these two units of measuring angles.

Degree to Radian Conversion

When converting from degrees to radians, it is essential to use the right conversion formula to maintain the accuracy of the measurement. The formula is as follows:
\[Radians = Degrees \times \frac{\pi}{180}\]
This formula derives from the fact that \(180^{\text{o}}\) is equivalent to \(\pi\) radians. Hence, to find the radian measure, we multiply the number of degrees by \(\pi\) and then divide by \(180\).

• For every \(180^{\text{o}}\), there is \(\pi\) radians.

Angle Conversion

Angle conversion is a vital skill in various fields of study and real-world applications. Beyond converting degrees to radians, there are other angle measures like gradians and turns. Understanding how to convert between these allows us to solve a wide range of mathematical problems.

For example, engineers may use different units for precision, while mathematicians prefer radians for the simplicity it brings to calculus operations. It's essential to remember key conversion ratios and to consider the sign of the angle when performing conversions, especially when dealing with negative angles.

Trigonometry

Trigonometry is the branch of mathematics that deals with relationships between side lengths and angles of triangles. It's especially known for its functions – sine, cosine, and tangent – which are fundamental in solving problems about triangles and modeling periodic phenomena.

Using radians in trigonometry is particularly useful when working with these functions in calculus, as they simplify the derivatives and integrals of trigonometric functions. The language of radians is essential when exploring the properties of periodic functions, making angle conversion a crucial skill for any student of mathematics.