Problem 5 - Precalculus

Question

Convert to radian measure. Express your answers both in terms of \(\pi\)and as decimal approximations rounded to two decimal places. (a) \(45^{\circ}\) (b) \(90^{\circ}\) (c) \(135^{\circ}\)

Step-by-Step Solution

Answer

(a) \(\frac{\pi}{4}\) or 0.79; (b) \(\frac{\pi}{2}\) or 1.57; (c) \(\frac{3\pi}{4}\) or 2.36.

Step 1: Understanding Degree to Radian Conversion

To convert degrees to radians, we use the conversion factor \( \frac{\pi}{180} \). This factor arises because \(180^{\circ} = \pi \text{ radians}\). Any angle in degrees can be converted to radians by multiplying it by \( \frac{\pi}{180} \).

Step 2: Calculating Radians for 45 Degrees

We begin by converting \(45^{\circ}\). Apply the conversion formula: \(45^{\circ} \times \frac{\pi}{180} = \frac{45\pi}{180} = \frac{\pi}{4}\). For a decimal approximation, \(\frac{\pi}{4} \approx 0.7854\), which rounds to 0.79.

Step 3: Calculating Radians for 90 Degrees

Next, convert \(90^{\circ}\). Using the conversion formula: \(90^{\circ} \times \frac{\pi}{180} = \frac{90\pi}{180} = \frac{\pi}{2}\). The decimal approximation is \(\frac{\pi}{2} \approx 1.5708\), which rounds to 1.57.

Step 4: Calculating Radians for 135 Degrees

Finally, convert \(135^{\circ}\). Using the conversion formula: \(135^{\circ} \times \frac{\pi}{180} = \frac{135\pi}{180} = \frac{3\pi}{4}\). The decimal approximation is \(\frac{3\pi}{4} \approx 2.3562\), which rounds to 2.36.

Key Concepts

Degree to Radian

Degrees and radians are both units for measuring angles; however, they are used in different contexts. Converting between them is a crucial skill in trigonometry. To transform an angle from degrees to radians, we multiply the degree measure by the fraction \( \frac{\pi}{180} \). This is because a full circle is \(360^{\circ}\) and is equivalent to \(2\pi\) radians, which means \(180^{\circ} = \pi\) radians. Therefore, the conversion factor \( \frac{\pi}{180} \) stems from equating these two angles.For example, to convert \(45^{\circ}\) to radians:

• Multiply \(45^{\circ}\) by \( \frac{\pi}{180} \)
• This results in \( \frac{45\pi}{180} = \frac{\pi}{4} \) radians

For accuracy and ease of calculation, degrees often convert to radians in terms of \(\pi\), emphasizing the mathematical constant's significance in calculations.

Angle Measurement

Understanding how angles are measured is fundamental in mathematics, especially in geometry and trigonometry. Angles describe the amount of turn between two lines around their common point, called the vertex. The two main units of angle measurement are:

• Degrees: A more intuitive measure, often used in everyday life. A full circle is \(360^{\circ}\).
• Radians: A more natural measure in mathematics, especially in calculus and trigonometry. A full circle is \(2\pi\) radians.

Radians provide a simpler solution in many mathematical contexts as they relate directly to the circle's radius. This connection creates more straightforward formulae in calculus and trigonometry, particularly in functions and derivatives.

Trigonometry Concepts

Trigonometry deals with the relationships between the angles and sides of triangles, especially right triangles. A critical aspect of trigonometry is understanding angle measurement and conversions, such as turning degrees into radians. When studying trigonometry, you'll encounter several essential functions including sine, cosine, and tangent. These functions are often more elegantly formulated in radian measure:

• Sine (sin): Relates opposite side over hypotenuse in a right triangle. Periodicity in radians helps in calculus.
• Cosine (cos): Relates adjacent side over hypotenuse. Symmetry properties are clearer in radians.
• Tangent (tan): Relates opposite over adjacent side. Tangent curve derivatives are simpler in radians.

In practice, converting degrees to radians simplifies trigonometric problem-solving and brings balance and harmony to equations.