Using Trigonometry to Determine Unknown Sides and Angles in Oblique Triangles

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Oblique triangles are triangles that do not contain a right angle. They can be either acute (all angles less than 90°) or obtuse (one angle greater than 90°). Determining the unknown sides and angles of these triangles requires the use of trigonometry, specifically the Law of Sines and the Law of Cosines.

Understanding Oblique Triangles

Unlike right triangles, oblique triangles do not have a 90° angle, so the basic Pythagorean theorem is not applicable. Instead, we rely on the Law of Sines and Law of Cosines to find missing information.

The Law of Sines

The Law of Sines relates the ratios of the sides of a triangle to the sines of their opposite angles. It is useful when we know either:

  • Two angles and one side (AAS or ASA), or
  • Two sides and a non-included angle (SSA)

The formula is:

(a / sin A) = (b / sin B) = (c / sin C)

The Law of Cosines

The Law of Cosines is used when we know:

  • Two sides and the included angle (SAS), or
  • All three sides (SSS)

The formula is:

c² = a² + b² – 2ab * cos C

Steps to Find Unknown Sides and Angles

To solve for unknown sides or angles in an oblique triangle, follow these steps:

  • Identify which law applies based on the given information.
  • Use the Law of Sines or Law of Cosines to set up an equation.
  • Solve for the unknown side or angle algebraically.
  • If necessary, repeat the process to find all missing parts.

Example Problem

Suppose you know two sides of an oblique triangle: a = 8 units, b = 10 units, and the included angle C = 60°. Find side c and angles A and B.

First, use the Law of Cosines to find side c:

c² = a² + b² – 2ab * cos C

c² = 8² + 10² – 2(8)(10) * cos 60°

c² = 64 + 100 – 160 * 0.5

c² = 164 – 80 = 84

c = √84 ≈ 9.17 units

Next, find angle A using the Law of Sines:

(a / sin A) = (c / sin C)

8 / sin A = 9.17 / sin 60°

sin A = (8 * sin 60°) / 9.17

sin A ≈ (8 * 0.866) / 9.17 ≈ 6.928 / 9.17 ≈ 0.756

A ≈ sin⁻¹(0.756) ≈ 49°

Finally, find angle B:

B = 180° – C – A = 180° – 60° – 49° = 71°

Conclusion

Using the Law of Sines and Law of Cosines, you can find unknown sides and angles in oblique triangles. Practice with different problems to strengthen your understanding of these important trigonometric tools.