Convert Angles Between Degrees and Radians - Online Calculator

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What Are Degrees and Radians?

Angles are measured in two primary units: degrees (\(\circ\)) and radians (rad).

Degrees: A degree is ¹⁄₃₆₀ of a full circle. This system is widely used in everyday contexts (e.g., navigation, construction).

Radians: A radian is the angle subtended at the center of a circle by an arc equal to the radius. One full circle is \(2 \pi\) radians (~6.283 rad). Radians are essential in calculus, physics, and advanced mathematics.

Key Relationship:

• \(360^\circ = 2 \pi \) radians
• \(180^\circ = \pi \) radians

Why Convert Between Degrees and Radians?

• Math & Science: Calculus, trigonometry, and physics formulas (e.g., angular velocity) require radians.
• Engineering: Rotational systems and signal processing often use radians.
• Programming: Languages like Python and MATLAB default to radians for trigonometric functions.

How to Convert Degrees to Radians

Formula: \(Radians = Degrees \times \pi / 180\)

Example: Convert \(45^ \circ\) to radians:

\(45 \times \pi / 180 =\pi /4 \) rad \(\approx \) 0.785 rad

Common Conversions:

• \(30^ \circ = \pi /6\) rad
• \(90^ \circ = \pi /2\) rad
• \(180^ \circ = \pi \) rad

How to Convert Radians to Degrees

Formula: \(Degrees = Radians \times 180 / \pi \)

Example: Convert \((3 \pi/4)\) rad to degrees:

\((3 \pi/4) \times 180 / \pi = 135^ \circ \)

Common Conversions:

• \( \pi/3 rad = 60^\circ \)
• \( \pi/6 rad = 30^\circ \)
• \( 2 \pi rad = 360^\circ \)

Visualizing the Relationship

Imagine slicing a pizza:

• Degrees: Each slice is labeled from \( 0^\circ \) to \( 360^\circ \).
• Radians: Each slice is labeled in terms of \( \pi \) (e.g., \(\pi/2\), \( \pi \)).

It's the same angle, just different "units"!

Tips for Avoiding Mistakes

• Always include \( \pi \) in degree-to-radian conversions.
• Check that your calculator is in the correct mode (DEG vs. RAD).
• Simplify fractions when possible: \( \pi/4 \) is cleaner than 0.785 rad.

Real-World Applications

• Physics: Angular displacement and motion.
• Computer Graphics: Rotation in 2D/3D rendering.
• Astronomy: Measuring apparent size and motion of celestial objects.

FAQs About Angle Conversions

Q: Why do mathematicians prefer radians?
A: Radians simplify calculus operations, such as derivatives of sine and cosine.

Q: Can I convert negative angles?
A: Yes! For example, \({-90}^\circ = -\pi/2\) radians.

Q: How can I estimate radians without \( \pi \)?
A: Use \( \pi \approx 3.14\) . So \( 60^\circ \approx 1.05\) radians.