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How Do You Solve Sine Rule?

Published in Trigonometry Calculations 3 mins read

The Sine Rule is solved by using the relationship between the sides and angles of any triangle. It states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant. This rule is particularly helpful when you know at least one side and its opposite angle, along with either another side or another angle in a non-right-angled triangle.

Understanding the Sine Rule

The sine rule can be expressed as follows:

  • sin(A) / a = sin(B) / b = sin(C) / c

    Where:

    • A, B, and C are the angles of the triangle.
    • a, b, and c are the lengths of the sides opposite those angles.

Steps to Solve using the Sine Rule

  1. Identify Knowns: Determine which angles and side lengths you know from the problem. You will need at least one pair of an angle and its opposite side, in addition to either another angle or side.

  2. Set Up Proportion: Select two ratios from the sine rule formula, including at least one ratio with both the angle and side known, and another with one known and one unknown.

    • For example, if you know angle A, side a, and angle B and you're trying to find side b, use:
      sin(A) / a = sin(B) / b
  3. Solve the Equation:

    • Cross-multiply to rearrange and isolate the unknown variable.
    • Use a calculator to find the sine values for the given angles.
    • Solve for the unknown length or angle.

Examples and Practical Insights

  • Finding a Missing Side:

    • If you have a triangle where:
      • Angle A = 40°, side a = 8cm, and angle B = 60°.
      • To find side b, you set up the equation:
        sin(40°) / 8 = sin(60°) / b
      • Cross multiply to get: b = (8 * sin(60°)) / sin(40°)
      • Solve for b: b ≈ 11.03 cm
  • Finding a Missing Angle:

    • If you have a triangle where:
      • Angle A = 30°, side a = 5cm and side b = 7cm
      • To find Angle B, you set up the equation:
        sin(30°) / 5 = sin(B) / 7
      • Rearranging the terms: sin(B) = (7 * sin(30°)) / 5
      • Solve for sin(B): sin(B) = 0.7
      • Apply the inverse sine (arcsin) to find B: B = arcsin(0.7) ≈ 44.4°

When to Use the Sine Rule

According to the reference provided ([Part of a video titled Maths Tutorial: Trigonometry Law of Sines / Sine Rule - YouTube]()), the sine rule is used when dealing with non-right-angled triangles, where you have:

  • Two angles and one side (AAS or ASA)
  • Two sides and one non-included angle (SSA) - Care is needed in this case as it can sometimes give two possible triangles.

Note: The video also points out the Sine Rule (or Law of Sines) is also called the "Law of Signs" (0:12). The cosine rule or law of cosines is another alternative.

Practical Applications

The sine rule is widely used in:

  • Navigation: Calculating distances and angles in map reading or nautical navigation.
  • Surveying: Determining lengths and angles in land surveys.
  • Engineering: Calculating forces in mechanical systems.
  • Physics: Determining directions and magnitudes of vectors.

Summary

To solve the sine rule: set up the ratios of the sine of each angle to the length of its opposite side. If you have sufficient information (as described above), you can solve for missing sides or angles in a non-right-angled triangle.