How to Use Law of Cosines to Solve for Unknown Sides in Non-right Triangles

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Understanding how to find unknown sides of non-right triangles is a crucial skill in geometry. The Law of Cosines provides a powerful method for solving these triangles when you know two sides and the included angle or all three sides.

What is the Law of Cosines?

The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. It is expressed as:

c2 = a2 + b2 – 2abcosC

where a and b are the known sides, C is the included angle, and c is the side opposite angle C.

Using the Law of Cosines to Find an Unknown Side

Suppose you have a triangle with sides a = 7 units, b = 10 units, and the included angle C = 60°. To find side c, substitute the known values into the formula:

c2 = 72 + 102 – 2(7)(10) cos 60°

Calculate each term:

  • 72 = 49
  • 102 = 100
  • 2(7)(10) = 140
  • cos 60° = 0.5

Now, substitute these into the equation:

c2 = 49 + 100 – 140 * 0.5

Simplify:

c2 = 149 – 70 = 79

Finally, take the square root to find c:

c ≈ √79 ≈ 8.89 units

Using the Law of Cosines to Find an Unknown Angle

If you know all three sides of a triangle, you can find an unknown angle. Rearranged, the Law of Cosines becomes:

cos C = (a2 + b2 – c2) / (2ab)

For example, if a = 8, b = 6, and c = 10, then:

cos C = (82 + 62 – 102) / (2 * 8 * 6)

Calculate numerator:

64 + 36 – 100 = 0

Calculate denominator:

2 * 8 * 6 = 96

Thus:

cos C = 0 / 96 = 0

Find the angle:

C = cos-1(0) = 90°

Conclusion

The Law of Cosines is an essential tool for solving non-right triangles. It helps you find unknown sides when you know two sides and the included angle or all three sides. Practice applying this law to various problems to strengthen your understanding of triangle geometry.