Math Formula for Degree Angle

Math Formula for Degree Angle

Understanding Degree Angles

A degree is a unit of measurement for angles, where a full rotation around a point is 360 degrees (°). Degree angles are commonly used in geometry, trigonometry, and engineering.

Basic Formulas for Degree Angles

Conversion Between Degrees and Radians

The relationship between degrees and radians is given by the formula:

\[ 1° = \frac{\pi}{180} \text{ radians} \]

To convert degrees to radians:

\[ \theta_{rad} = \theta_{deg} \times \frac{\pi}{180} \]

To convert radians to degrees:

\[ \theta_{deg} = \theta_{rad} \times \frac{180}{\pi} \]

Sum of Interior Angles of a Polygon

The sum of the interior angles of an \( n \)-sided polygon is:

\[ S = (n - 2) \times 180° \]

Exterior Angles of a Polygon

The sum of the exterior angles of any polygon is always 360°:

\[ S_{ext} = 360° \]

Complementary and Supplementary Angles

Two angles are complementary if their sum is 90°:

\[ \alpha + \beta = 90° \]

Two angles are supplementary if their sum is 180°:

\[ \alpha + \beta = 180° \]

Trigonometric Functions in Degrees

Basic Trigonometric Ratios

For a right triangle, the trigonometric functions in degrees are:

\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]

\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \]

\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]

Common Angle Values

Angle (°)	sin(θ)	cos(θ)	tan(θ)
0°	0	1	0
30°	1/2	\( \sqrt{3}/2 \)	\( \sqrt{3}/3 \)
45°	\( \sqrt{2}/2 \)	\( \sqrt{2}/2 \)	1
60°	\( \sqrt{3}/2 \)	1/2	\( \sqrt{3} \)
90°	1	0	Undefined

Applications in Science and Engineering

Angles are used in physics for motion analysis, optics, and wave mechanics. Engineers use degree measurements for structural analysis and robotics.

Advanced Angle Properties

Law of Sines

For any triangle:

\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]

Law of Cosines

\[ c^2 = a^2 + b^2 - 2ab \cos C \]

Examples

Example 4: Finding an Angle Using the Law of Sines

If \( A = 40° \), \( a = 10 \), and \( b = 15 \), find \( B \).

\[ \frac{10}{\sin 40°} = \frac{15}{\sin B} \]

Solving for \( B \):

\[ B = \sin^{-1}(\frac{15 \sin 40°}{10}) \]

Example 5: Calculating Distance Using an Angle

A lighthouse beam extends 15 km at an angle of 30°. How high is the beam above the water?

Using \( h = 15 \sin 30° \), we find:

\[ h = 15 \times 0.5 = 7.5 \text{ km} \]

Conclusion

Mastering degree angles and their applications is vital in science, engineering, and everyday problem-solving. Understanding their properties allows us to analyze and interpret spatial relationships effectively.

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