Angles and Their Measurement

Degrees and Radians

Angles can be measured in degrees or radians. Understanding how to convert between these units is fundamental in trigonometry.

• Degree: A degree is 1/360 of a full rotation.

• Radian: A radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.

• Conversion Formula: To convert radians to degrees, use the formula:

• Example: Convert radians to degrees:

• Example: Convert radians to degrees:

Evaluating Trigonometric Functions

Exact Values of Trigonometric Functions

Trigonometric functions can be evaluated for specific angles, often using the unit circle or known values.

• Cosine:

  • On the unit circle, is in the second quadrant.

• Cotangent:

• Secant:

  • , so

Applications of Trigonometry

Solving Real-World Problems

Trigonometry is used to solve problems involving distances, heights, bearings, and circular motion.

• Bearings and Navigation:

  • Problems may involve finding the distance between two points given bearings and distances.

  • Law of Cosines: Used when two sides and the included angle are known:

  • Example: Radar stations A and B are 8.6 km apart. Station A detects a plane at a bearing of , and station B detects the same plane at a bearing of . Find the distance from B to C using the Law of Sines or Cosines.

• Height Problems:

  • Given angles of elevation and horizontal distances, use tangent function to find heights.

  • Example: From a point, the angle of elevation to a building is . After walking back 35 ft, the angle is . Find the height :

    • Let be the original distance. ,

• Circular Motion and Arc Length:

  • Arc Length Formula: , where is arc length, is radius, and is angle in radians.

  • Example: A rope is wound around a drum of radius 0.327 m through an angle of .

    • Convert angle to radians:

    • Calculate arc length:

• Gear Rotation:

  • When two gears are meshed, the arc length traveled by each is equal at the point of contact.

  • Formula:

  • Example: If a smaller gear of radius 12.5 cm rotates through , find the angle the larger gear of radius 34.4 cm rotates through:

    • Solve for

Summary Table: Key Trigonometric Formulas

Problem-Solving Steps in Trigonometry

General Approach

• Draw a figure and label all sides and angles.

• Show original substitution into the formula.

• Solve the equation algebraically.

• State the final simplified answer with label and unit.

Additional info:

• Some problems require rounding to one decimal place and labeling units (e.g., meters, degrees).

• Always show work, including diagrams, for full credit.