Notes on Law of Sines and Cosines

Law of Sines and Cosines

Overview

  • Important rules in trigonometry for solving triangles.

  • Particularly useful for calculating unknown sides and angles in various triangle scenarios.

Law of Sines

  • Applicable when given:

    • ASA (Angle-Side-Angle)

    • AAS (Angle-Angle-Side)

    • SSA (Side-Side-Angle)

  • Formula: ( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} )

  • Can find all missing sides and angles with the Law of Sines.

  • Example solving triangles (keep answers rounded to 3 decimal places).

Law of Cosines

  • Used when given:

    • SAS (Side-Angle-Side)

    • SSS (Side-Side-Side)

  • Standard Form: ( a^2 = b^2 + c^2 - 2bc \cos A )

  • Alternative Form available. Useful for calculating angles or sides not opposite the largest angle.

  • Example solving triangles using the Law of Cosines.

Cases in SSA

  1. One Solution Case:

    • If side opposite the given angle is larger, only one triangle is possible.

  2. No Solution Case:

    • The angle is obtuse, or the side opposite is smaller than the other provided side.

    • Check if the angle is acute and the sine of the missing angle exceeds 1.

  3. Two Solutions Case:

    • If the angle is acute and smaller than the given side, calculate for both triangles.

    • Solutions referred to as B and B' for the two angles.

Triangle Examples

  1. Example - Solve a triangle with:

    • Given: A, C, and a side (b)

    • Calculate remaining angles and sides accordingly.

  2. Trigonometric Applications:

    • Example calculations involving real-life scenarios.

    • Boat Race: Distance from A to B to C and returning to A.

    • Plane Flight: Calculate distance between cities with angles.

Area of Oblique Triangle

  • Area formulas:

    • ( Area = \frac{1}{2}ab \sin C ) (Given SAS)

    • ( Area = \frac{1}{2}bc \sin A )

    • ( Area = \frac{1}{2}ac \sin B )

Heron’s Formula for Area of Triangle

  • Area = ( \sqrt{s(s-a)(s-b)(s-c)} )

  • Where ( s = \frac{a+b+c}{2} )

Example Calculation for Area

  • Given lengths of sides: a = 5, b = 7, c = 10

  • Calculate semi-perimeter and area using Heron’s formula.

  • Round to the nearest thousandth.

  • Conclusion: Understanding and applying Law of Sines and Law of Cosines is essential for solving triangle-related problems in trigonometry and geometry.