Problem 16 - Precalculus

Question

Convert from degrees to radians. Leave the answers in terms of \(\pi\). $$90^{\circ}$$

Step-by-Step Solution

Answer

The angle is \(\frac{\pi}{2}\) radians.

Step 1: Understand the Conversion Formula

To convert an angle from degrees to radians, use the conversion formula: \[ ext{Radians} = ext{Degrees} \times \frac{\pi}{180^{ ext{o}}}\]This formula helps change an angle given in degrees to radians.

Step 2: Substitute the Given Angle

Substitute the given angle, which is \(90^{\circ}\), into the conversion formula. This gives:\[ ext{Radians} = 90^{\circ} \times \frac{\pi}{180^{\circ}}\]

Step 3: Simplify the Expression

Simplify the fraction \(\frac{90}{180}\) which simplifies to \(\frac{1}{2}\). This makes the expression:\[\frac{1}{2}\pi\]Thus, the angle in radians is \(\frac{\pi}{2}\).

Key Concepts

Angle Measurement

Angles are essential in mathematics for describing the turn between two lines. They can be measured in two main units: degrees and radians.
Degrees are more commonly used in everyday situations and schooling. A full circle is 360 degrees (\(360^{\circ}\)), so a right angle is 90 degrees (\(90^{\circ}\)).
Radians, on the other hand, are the standard unit of angular measure used in many mathematical disciplines, including calculus and trigonometry.
This unit correlates directly to the circle's radius. For instance, a full circle is equal to \(2\pi\) radians.

• Radians measure the actual distance along the arc of a circle, which makes them more intuitive in higher mathematics.
• This connection to the radius of the circle helps simplify formulas for mathematical analysis.

Understanding both degree and radian measurement is crucial for mastering conversion between them and applying mathematics in different contexts.

Conversion Formula

When converting angles from degrees to radians, remember the essential conversion formula:
\[\text{Radians} = \text{Degrees} \times \frac{\pi}{180^{\circ}}\]
This formula ensures that the angle's measurement is consistent within mathematical calculations that require radians. Here's how you use it:

• Identify the angle in degrees, which serves as your starting point.
• Multiply the degree measure by \(\frac{\pi}{180}\) to get the angle in radians.

Using this formula will systematically allow any angle given in degrees to be represented in radians, which is crucial for trigonometric analysis.
For example, converting \(90^{\circ}\):

• Substitute the value in the formula: \(90 \times \frac{\pi}{180}\).
• Simplify to get \(\frac{1}{2}\pi\).

This means that \(90^{\circ}\) in radians is \(\frac{\pi}{2}\), a conversion extensively useful in mathematics.

Trigonometric Functions

Trigonometric functions are foundational in mathematics, involving the relationships between the angles and sides of triangles.
They also play a significant role in modeling waves, oscillations, and other periodic phenomena.
To efficiently use trigonometric functions, understanding angles in radians is advantageous, since it simplifies computations.

• Sine, cosine, and tangent are common trigonometric functions that rely on angle measures.
• These functions often have simpler derivatives and integrals when angles are measured in radians.
• Radians make the functions periodic with nicely predictable results.

Consider a trigonometric function like sine:

• \(\sin(\frac{\pi}{2})\), which equals 1.
• In radians, this value is straightforward since each angle in radians corresponds naturally to the unit circle.

Mastering the transition from degrees to radians makes working with trigonometric functions not only more efficient but also deepens the understanding of these essential mathematical tools.