Problem 66 - Algebra and Trigonometry Real Mathematics, Real People

Question

Rewrite each angle in radian measure as a multiple of \(\pi .\) (Do not use a calculator.) (a) \(-330^{\circ}\) (b) \(144^{\circ}\)

Step-by-Step Solution

Answer

(a) \(-11/3 \pi\) (b) \(4/5 \pi\)

Step 1: Conversion of \(-330^{\circ}\)

Apply the conversion factor \(180° = \pi\) radians to \(-330^{\circ}\). The conversion involves multiplying \(-330°\) by \(\pi/180\). Hence, the calculations will look as follows, \(-330° × \pi/180 = -11/3 \pi\).

Step 2: Conversion of \(144^{\circ}\)

Similarly, multiply \(144^{\circ}\) by \( \pi/180\). This will give \(144^{\circ} × \pi/180 = 4/5 \pi\).

Key Concepts

Angle Conversion

Angles are an essential part of trigonometry and involve measuring the amount of rotation from a starting point, usually in different units like degrees and radians. Converting angles means transforming their representation from one unit to another. This is particularly important in trigonometry, where radians are often used because they provide a direct link between angle measure and arc length. A full circle is usually represented as 360 degrees or \(2\pi\) radians. Therefore, understanding how to transition between degrees and radians helps in comprehending trigonometric functions more naturally.

When you convert an angle, you maintain its value but change its unit representation. This involves using a specific relation known as the conversion factor, which provides the necessary multiplicative transformation between degrees and radians. This knowledge is crucial because it enhances computational efficiency and understanding in mathematical and practical applications.

Degrees to Radians

The conversion from degrees to radians is an arithmetic operation that ensures angles are expressed in terms of radians, a unit more commonly utilized in higher mathematics and physics. To convert an angle from degrees to radians, you use the fundamental relationship: 180 degrees is equivalent to \(\pi\) radians. This is the conversion factor that ties the two units together.

Here's a simple process to follow:

• Multiply the angle in degrees by \(\pi/180\).
• Simplify the expression to get the angle in radians.

For example, to convert \(-330^{\circ}\) to radians, multiply by \(\pi/180\):
\[-330 \times \frac{\pi}{180} = -\frac{11\pi}{3}\]
Similarly, for \(144^{\circ}\):
\[144 \times \frac{\pi}{180} = \frac{4\pi}{5}\]
This final expression tells us the angle in terms of \(\pi\), which is often more convenient for advanced calculations in trigonometry.

Trigonometric Conversion

Understanding trigonometric conversion is about more than just changing units; it involves adapting trigonometric expressions to suit the specific context of a problem. Radians become essential because they simplify the expressions of trigonometric identities and functions. This conversion is helpful not only for theoretical purposes but also for solving real-world problems where these functions model periodic phenomena such as waves or oscillations.

Here’s why radians matter in trigonometric conversion:

• The radian measure connects directly with the unit circle — a crucial concept in trigonometry that aids in defining functions such as sine and cosine.
• Using radians can make differentiation and integration of trigonometric functions more straightforward.

So, when you convert degrees to radians, you're not just changing a number; you are enabling more precise and often simpler mathematical modeling. This ultimately helps in improving problem-solving efficiency, especially in fields like calculus, physics, and engineering.