Question

Convert the following angles in radians to degrees: (a) \(0.3609\) (b) \(0.4771\) (c) \(1.3692\) (d) \(\frac{\pi}{3}\) (e) \(\frac{2 \pi}{3}\) (f) \(6 \pi\) (g) \(\frac{\pi}{5}\) (h) \(\frac{3 \pi}{2}\)

Short answer

Question: Convert the following angles from radians to degrees: (a) 0.3609 radians (b) 0.4771 radians (c) 1.3692 radians (d) π/3 radians (e) 2π/3 radians (f) 6π radians (g) π/5 radians (h) 3π/2 radians Answer: (a) ≈ 20.663° (b) ≈ 27.344° (c) ≈ 78.486° (d) 60° (e) 120° (f) 1080° (g) 36° (h) 270°

Step-by-step solution

(a) Convert 0.3609 radians to degrees

Using the conversion formula, we have: Degrees = (0.3609 × 180) ÷ π Using a calculator, find the value: Degrees ≈ 20.663°

(b) Convert 0.4771 radians to degrees

Using the conversion formula, we have: Degrees = (0.4771 × 180) ÷ π Using a calculator, find the value: Degrees ≈ 27.344°

(c) Convert 1.3692 radians to degrees

Using the conversion formula, we have: Degrees = (1.3692 × 180) ÷ π Using a calculator, find the value: Degrees ≈ 78.486°

(d) Convert π/3 radians to degrees

Using the conversion formula, we have: Degrees = (π/3 × 180) ÷ π Simplify: Degrees = 60°

(e) Convert 2π/3 radians to degrees

Using the conversion formula, we have: Degrees = (2π/3 × 180) ÷ π Simplify: Degrees = 120°

(f) Convert 6π radians to degrees

Using the conversion formula, we have: Degrees = (6π × 180) ÷ π Simplify: Degrees = 1080°

(g) Convert π/5 radians to degrees

Using the conversion formula, we have: Degrees = (π/5 × 180) ÷ π Simplify: Degrees = 36°

(h) Convert 3π/2 radians to degrees

Using the conversion formula, we have: Degrees = (3π/2 × 180) ÷ π Simplify: Degrees = 270°

Key concepts

Radians to Degrees

Converting radians to degrees involves a straightforward process based on a known mathematical constant. Radians and degrees are both units used for measuring angles, but they differ in their application. Radians are often used in calculus, while degrees are favored in general geometry. To convert radians into degrees, we use the conversion formula: \[ \text{Degrees} = \left( \text{Radians} \times \frac{180}{\pi} \right) \] This formula arises because a full circle is \(360\) degrees, which is equivalent to \(2\pi\) radians. Thus, dividing \(360\) by \(2\pi\) gives you the conversion factor \(\frac{180}{\pi}\). This operation essentially scales the radian measure to the degree scale. When applying this formula, always ensure your calculator is set to radians when performing calculations with radian values to achieve accurate results.

Trigonometry

Trigonometry is a branch of mathematics that focuses on the relationships between the angles and sides of triangles. It's a vital discipline for anyone studying geometry, calculus, and various fields of engineering and physics. Understanding trigonometry often requires familiarity with different angle units, including degrees and radians. In trigonometry, angles can be expressed in either radians or degrees, depending on what is practiced in the field or required by the particular application. Despite the differences in units, the fundamental relationships between the functions remain the same. This is crucial when working with trigonometric functions like sine, cosine, and tangent, which rely on angle inputs. Both unit systems are utilized extensively:

• Degrees are common for non-technical applications and simpler geometry problems.
• Radians are necessary for more advanced calculus and functions that describe periodic phenomena like waves.

Choosing the correct unit often depends on the context of the problem and the preferences of the mathematical field.

Mathematical Formulas

Mathematical formulas are the language through which complex ideas and calculations are communicated. They provide a universal method for describing relationships between numbers and variables. Formulas are crucial for solving equations, converting measurements, and understanding various mathematical principles. In the context of angle conversion, the formula: \[ \text{Degrees} = \left( \text{Radians} \times \frac{180}{\pi} \right) \] is not just a simple conversion but a representation of the underlying relationship between radians and degrees. This intrinsic connection is fundamental to many areas of mathematics, particularly when working with circular functions and sinusoidal models. Key elements to remember when using formulas:

• Always use the correct formula for your calculations.
• Understand the relationship each formula expresses.
• Check the units you are using to ensure accuracy.

Using formulas correctly is essential for anyone looking to excel in mathematics and is considered a fundamental skill in any scientific endeavor.