Law of Sines

Law of Sines

Introduction

  • The law of sines is used to solve triangles that are not right triangles.
  • It provides a relationship between the sides and angles of any triangle.

Convention

  • Angles are labeled as A, B, and C.
  • Sides opposite to these angles are labeled as a, b, and c, respectively.

Derivation of the Law of Sines

  1. Altitude: An altitude (height (h)) is dropped from one vertex perpendicular to the opposite side.

  2. Right Triangles: This creates two right triangles within the original triangle.

  3. Sine Definition: Using the sine function (SOHCAHTOA):

    • In one right triangle: sin⁡(A)=hb\sin(A) = \frac{h}{b}
    • In the other right triangle: sin⁡(B)=ha\sin(B) = \frac{h}{a}

  4. Solving for h:

    • \h = b \sin(A)
    • \h = a \sin(B)

  5. Equating the expressions for h: Since (h) is the same in both equations:

    • \b \sin(A) = a \sin(B)

  6. Law of Sines Formula: Dividing both sides by (\sin(A)) and (\sin(B)) yields:

    • asin⁡(A)=bsin⁡(B)\frac{a}{\sin(A)} = \frac{b}{\sin(B)}

  7. Extending the Law: Including the third side and angle:

    • asin⁡(A)=bsin⁡(B)=csin⁡(C)\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}

Applications of the Law of Sines

Finding a Missing Side
  • Example: Given angle A = 55°, side a = 10, and angle B = 42°, find side b (x).
  • Applying the Law of Sines:
    • 10sin⁡(55°)=xsin⁡(42°)\frac{10}{\sin(55°)} = \frac{x}{\sin(42°)}
  • Solving for x:
    • x=10⋅sin⁡(42°)sin⁡(55°)x = \frac{10 \cdot \sin(42°)}{\sin(55°)}
    • x≈8.2x ≈ 8.2
  • Check: The larger side is opposite the larger angle.
Finding a Missing Angle
  • Example: Given side a = 8, angle A = 61°, and side b = 10, find angle B (x).
  • Applying the Law of Sines:
    • 8sin⁡(61°)=10sin⁡(x)\frac{8}{\sin(61°)} = \frac{10}{\sin(x)}
  • Solving for (\sin(x)):
    • sin⁡(x)=10⋅sin⁡(61°)8\sin(x) = \frac{10 \cdot \sin(61°)}{8}
  • Finding x:
    • x=arcsin⁡(8⋅sin⁡(61°)10)x = \arcsin(\frac{8 \cdot \sin(61°)}{10})
    • x≈44.4°x ≈ 44.4°

The Ambiguous Case (Angle-Side-Side)

  • Problem: When given an angle, a side, and another side (ASS), there might be two possible triangles.
  • Example: Given a 30° angle, a side of 10, and another side of 6.
    • The side of length 6 can be placed in two different positions, creating two different triangles.
  • Explanation: Consider a triangle with a 30-degree angle and a side of 10. The height (h) to this side from the opposite vertex is:
    • h=10sin⁡(30°)=5h = 10 \sin(30°) = 5
    • If the given side is a little bigger than 5, there are 2 possible triangles. However, if it's bigger than 10 there is only 1.

Dealing with the Ambiguous Case

  1. Check the Diagram: Visualize or draw the possible triangles.
  2. Obtuse Angle Consideration: The inverse sine function only returns angles between -90° and 90°.
    • If you expect an obtuse angle, subtract the calculator's result from 180°.
    • Obtuse Angle=180°−Acute Angle from Calculator\text{Obtuse Angle} = 180° - \text{Acute Angle from Calculator}

Inverse Sine Function Limitation

  • The inverse sine function (arcsin) gives angles in the first or fourth quadrant (between -90 and 90 degrees).
  • If the angle is obtuse (greater than 90 degrees), the calculator will not directly give you that angle. You need to subtract the result from 180 degrees to find the obtuse angle with the same sine value.