Bearings in CSEC Mathematics

Introduction

• Bearings are an application of trigonometry including sine rule and cosine rule.
• Cardinal points: North, South, West, and East.
• North is the most important direction and is always upwards on a diagram.

Definition of Bearings

• Bearings give a direction in terms of an angle from one point to another.
• Measured as the angle in the clockwise direction from the north line.
• Alternate angles are used to calculate bearings.

Calculating Bearings

• Bearings are represented using degrees (e.g., 115 degrees).
• Bearings must be expressed using three digits (e.g., 035 degrees for 35 degrees).

Applications

• Involves sine rule and cosine rule:
  • Sine Rule: Used when you have two sides and two angles.
  • Cosine Rule: Used when you have three sides and one angle.
• Can also involve Pythagoras' Theorem for right-angled triangles.

Example 1

• Given: Points P, Q, R with angles and distances.
• Tasks:
  • Calculate distance QR using cosine rule.
  • Calculate the area of triangle PQR using the formula: ( \frac{1}{2}ab\sin C ).
  • Find the bearing of R from P using the given information.

Step-by-step Calculation

1. Distance QR:

   • Use cosine rule due to three sides and one angle.
   • Calculate using given values, find QR.

2. Area of Triangle PQR:

   • Use area formula for a triangle with given sides and included angle.

3. Bearing Calculation:

   • Use alternate angles to calculate missing angles.
   • Add angles to find full bearing.

Example 2

• Given: Points P, Q, R with given bearings and distances.
• Tasks:
  • Calculate value of X using straight-line sum of angles.
  • Calculate distance RP using cosine rule due to involvement of three sides and one angle.
  • Determine the bearing of R from P using sine rule for angles.

Step-by-step Calculation

1. Calculate X:

   • Sum of angles in a straight line equals 180 degrees.

2. Distance RP:

   • Use cosine rule, calculate using given values.

3. Bearing Calculation:

   • Use sine rule to find internal angle and then calculate full bearing from given data.

Conclusion

• Bearings integrate sine rule and cosine rule heavily.
• Essential to understand trigonometry basics to solve bearings problems.
• Analyze questions carefully to understand what is required; interpret given information for bearing calculations.

Key Takeaway

• Thorough understanding of trigonometry rules is essential for solving bearing problems in CSEC Mathematics.