Law of Sines and Cosines

Introduction to the Law of Sines

Mastering the law of sines is essential for solving triangles, particularly non-right triangles. The law of sines relates the lengths of the sides of a triangle to the sines of its angles. It is especially useful in situations where we have either two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA).

The formula for the law of sines is:

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

Here, \(a\), \(b\), and \(c\) represent the sides of the triangle, while \(A\), \(B\), and \(C\) are the angles opposite these sides, respectively. This law allows for the determination of unknown sides or angles when certain other measurements are known.

Introduction to the Law of Cosines

The law of cosines is another critical tool for solving triangles, particularly when dealing with non-right triangles. It is an extension of the Pythagorean theorem and is useful for cases where two sides and the included angle are known (SAS) or when all three sides are known (SSS).

The formula for the law of cosines is:

\(c^2 = a^2 + b^2 - 2ab \cdot \cos C\)

Similarly, we can rearrange the formula for the other sides:

\(a^2 = b^2 + c^2 - 2bc \cdot \cos A\)
\(b^2 = a^2 + c^2 - 2ac \cdot \cos B\)

Applications in Solving Triangles

The laws of sines and cosines are invaluable for solving triangles in various fields, such as physics, engineering, and computer graphics. They allow for the calculation of unknown sides or angles, making them essential for determining distances, angles, and other geometric properties.

In practical applications, these laws help in navigation, architecture, and even in determining the position of celestial bodies. The choice between using the law of sines or cosines depends on the given information and what needs to be determined.

Worked Examples of Sines and Cosines

  \(C = 180^\circ - A - B = 180^\circ - 45^\circ - 60^\circ = 75^\circ\)
  

  \(\frac{a}{\sin A} = \frac{b}{\sin B}\)
  \(\frac{7}{\sin 45^\circ} = \frac{b}{\sin 60^\circ}\)
  \(\frac{7}{0.7071} = \frac{b}{0.8660}\)
  \(b = \frac{7 \times 0.8660}{0.7071} \approx 8.57\)
  

  \(c^2 = a^2 + b^2 - 2ab \cdot \cos C\)
  \(c^2 = 8^2 + 6^2 - 2 \times 8 \times 6 \times \cos 60^\circ\)
  \(c^2 = 64 + 36 - 96 \times 0.5\)
  \(c^2 = 100 - 48\)
  \(c^2 = 52\)
  \(c = \sqrt{52} \approx 7.21\)
  

Common Mistakes and How to Avoid Them

• Incorrect Angle Calculation: Always ensure the sum of angles in a triangle is \(180^\circ\). Double-check your angle values when using the law of sines.
• Misapplication of Laws: Use the law of sines for AAS, ASA, or SSA cases and the law of cosines for SAS or SSS cases.
• Rounding Errors: Be cautious with rounding intermediate steps, particularly when calculating trigonometric functions.

Practice Problems

1. Given a triangle with sides \(a = 5\), \(b = 7\), and angle \(C = 45^\circ\), find the length of side \(c\).
Show Solution

  \(c^2 = a^2 + b^2 - 2ab \cdot \cos C\)
  \(c^2 = 5^2 + 7^2 - 2 \times 5 \times 7 \times \cos 45^\circ\)
  \(c^2 = 25 + 49 - 70 \times 0.7071\)
  \(c^2 = 74 - 49.497\)
  \(c^2 = 24.503\)
  \(c = \sqrt{24.503} \approx 4.95\)
  

2. In a triangle, \(A = 30^\circ\), \(B = 45^\circ\), and \(a = 10\). Find side \(b\).
Show Solution

  \(C = 180^\circ - A - B = 180^\circ - 30^\circ - 45^\circ = 105^\circ\)
  \(\frac{a}{\sin A} = \frac{b}{\sin B}\)
  \(\frac{10}{\sin 30^\circ} = \frac{b}{\sin 45^\circ}\)
  \(\frac{10}{0.5} = \frac{b}{0.7071}\)
  \(b = \frac{10 \times 0.7071}{0.5} \approx 14.14\)
  

3. Find angle \(A\) in a triangle where \(a = 9\), \(b = 12\), and \(c = 15\).
Show Solution

  Use the law of cosines:
  \(a^2 = b^2 + c^2 - 2bc \cdot \cos A\)
  \(9^2 = 12^2 + 15^2 - 2 \times 12 \times 15 \cdot \cos A\)
  \(81 = 144 + 225 - 360 \cdot \cos A\)
  \(-288 = -360 \cdot \cos A\)
  \(\cos A = \frac{288}{360} = 0.8\)
  \(A = \cos^{-1}(0.8) \approx 36.87^\circ\)
  

See Also

• Graphs of Trig Functions