MATH04 Pre-Calculus Course Notes - Law of Sines

MATH04: Pre-Calculus

Course Outcome 4

Lesson 3: Law of Sines

Institution: MAPÚA UNIVERSITY

Overview

  • Applications of Trigonometry:

    • Trigonometry applies to oblique triangles (triangles without a 90° angle).

    • Example Scenario:

    • A trolley carries passengers from ground level to a mountain chateau.

    • Problem: Find the approximate length of the cable holding the trolley, given:

      • Angle A = 23°

      • Ground distance (horizontal) = 2000 m

      • Angle B = 67°.

Solving Triangles

  • Definition of Solving a Triangle:

    • Involves finding all sides' lengths and angles' measures of the triangle.

    • For oblique triangles, two main laws are utilized:

    • Law of Sines

    • Law of Cosines

Definitions

  • Law of Sines:

    • Used for oblique triangles when either:

    • Two angles and a side are known

    • Two sides and an angle opposite one of the sides are known

  • Law of Cosines:

    • Used for oblique triangles when either:

    • Two sides and the included angle are known

    • Three sides are known

Cases of Oblique Triangle Solutions

  • Categories:

    1. Two angles and one side are given.

    2. Two sides and an angle opposite one of the sides are given.

    3. Two sides and the angle between those sides are given.

    4. Three sides are given.

  • Notes:

    • Law of Sines applies to case 1 and 2.

    • Law of Cosines applies to case 3 and 4.

Solution Using Law of Sines

  • Consider Triangle ABC:

    • Let sides be denoted as a, b, and c opposite angles A, B, and C respectively.

    • If an altitude h is drawn to the base, the following relationships can be established:

    • rachextsin(B)=racbextsin(A)rac{h}{ ext{sin}(B)} = rac{b}{ ext{sin}(A)}

    • h=aracextsin(B)bh = a rac{ ext{sin}(B)}{b}

    • h=bracextsin(A)ah = b rac{ ext{sin}(A)}{a}

Derivation of Law of Sines

  • Start with:

    • bextsin(A)=aextsin(B)b ext{sin}(A) = a ext{sin}(B)

  • Rearranging gives:

    • racbextsin(A)aextsin(B)=1rac{b ext{sin}(A)}{a ext{sin}(B)} = 1

  • Thus, it can be expressed as:

    • racaextsin(A)=racbextsin(B)=raccextsin(C)rac{a}{ ext{sin}(A)} = rac{b}{ ext{sin}(B)} = rac{c}{ ext{sin}(C)}

Solving Oblique Triangles - Case I

Case I: Two Angles and One Side are Given

  1. Given measures:

    • Angle A = 51.3°

    • Angle B = 48.7°

    • Side a = 24.5 units

  2. Finding Side b:

    • By Sine Law:

      • racaextsin(A)=racbextsin(B)rac{a}{ ext{sin}(A)} = rac{b}{ ext{sin}(B)}

      • b=racaextsin(B)extsin(A)b = rac{a ext{sin}(B)}{ ext{sin}(A)}

      • Calculations:

      • b=24.5racextsin(48.7°)extsin(51.3°)b = 24.5 rac{ ext{sin}(48.7°)}{ ext{sin}(51.3°)}

      • Result: bext≈23.58extunitsb ext{≈ } 23.58 ext{ units}

Finding Angle C

  • Using sum of angles in a triangle:

    • C+A+B=180°C + A + B = 180°

    • C=180°−51.3°−48.7°=80°C = 180° - 51.3° - 48.7° = 80°

Finding Side c

  • Using Sine Law:

    • c=racaextsin(C)extsin(A)c = rac{a ext{sin}(C)}{ ext{sin}(A)}

    • Inserting values:

    • c=24.5racextsin(80°)extsin(51.3°)c = 24.5 rac{ ext{sin}(80°)}{ ext{sin}(51.3°)}

    • Result: cext≈30.92extunitsc ext{≈ } 30.92 ext{ units}

Solving Oblique Triangles - Case II

Case II: Two Sides and the Opposite Angle are Given (Ambiguous Case)

  • Problem: When provided two sides and the opposite angle, the triangle formation is variable (either none, one, or two possible triangles).

Example Details

  1. Evaluating Triangle ABC: Given values:

    • A = 67°

    • a = 18, b = 20

  2. Using Sine Law:

    • Find angle B:

    • racaextsin(A)=racbextsin(B)rac{a}{ ext{sin}(A)} = rac{b}{ ext{sin}(B)}

    • Result of angle B calculation may yield:

      • If ext{sin}(B) > 1, then no triangle possible (error).

Applications of Ambiguous Case

  • Example Calculation for Venus and Earth:

    • Assumption of 18° between respective positions of Venus and the Sun.

    • Distances: 93 million miles (from Earth to the Sun), 67 million miles (from Venus to the Sun).

  • Computation Method:

    • Identifying shortest and farthest positions between objects using established formulas for triangular distance.

Summary and Implications

  • Law of Sines is essential in determining oblique triangle side lengths and angles, particularly in geometric and real-world scenarios, such as navigation and engineering applications.

  • Ambiguous cases provide a unique challenge that requires careful analysis and validation of solutions.

Conclusion

  • Mastery of these concepts in trigonometry lays foundational knowledge crucial for advanced mathematical applications in various fields.

  • Further exploration and practice problems are encouraged to hone proficiency in these skills.