Introduction to Oblique Triangles and the Law of Cosines

In trigonometry, triangles are classified into two broad categories: right triangles and oblique triangles. Oblique triangles—those that contain no right angle (90°)—pose a greater challenge when it comes to solving for unknown sides or angles. While the Pythagorean theorem and basic trigonometric ratios (sine, cosine, tangent) work seamlessly for right triangles, they are insufficient for oblique triangles. To solve these triangles, we need the Law of Sines and the Law of Cosines.

The Law of Cosines is particularly effective when we know two sides of an oblique triangle and the included angle (SAS case) or when we know all three sides (SSS case) and want to find an angle. It is a generalization of the Pythagorean theorem: for a triangle with sides a, b, and c, and angle C opposite side c, the formula is:

c² = a² + b² – 2ab cos(C)

The same pattern applies cyclically for the other sides: a² = b² + c² – 2bc cos(A) and b² = a² + c² – 2ac cos(B). This set of equations is flexible and reliable for any triangle—whether acute, obtuse, or right.

In this article, we’ll focus on using the Law of Cosines to find unknown sides when we are given two sides and the included angle. We’ll work through step‑by‑step examples, offer practical tips for avoiding common mistakes, and link to additional resources for deeper study.

Understanding the Law of Cosines: Formula and Derivation

Why Is It Called the “Law of Cosines”?

The name comes from the appearance of the cosine function in the correction term –2ab cos(C). If the included angle is 90°, then cos(90°) = 0, and the formula reduces to the Pythagorean theorem: c² = a² + b². For other angles, we “adjust” the sum of squares by a factor that accounts for the projection of one side onto the direction of the other.

The General Formula for All Sides

To apply the Law of Cosines confidently, memorize the three symmetrical forms:

  • For side c: c² = a² + b² – 2ab cos(C)
  • For side a: a² = b² + c² – 2bc cos(A)
  • For side b: b² = a² + c² – 2ac cos(B)

Notice that the side we want to find is always on the left side, and the angle used in the cosine term is opposite that side. The two known sides are the ones adjacent to the included angle.

Derivation in Brief

The law can be derived by dropping a perpendicular from one vertex to the opposite side and applying the Pythagorean theorem to the resulting right triangles. Alternatively, it can be proven using vector geometry: the square of the magnitude of a vector difference equals the sum of squares of the magnitudes minus twice the dot product. The dot product introduces the cosine of the angle between the vectors.

Understanding the derivation gives you confidence that the formula always works—for any triangle shape.

Applying the Law of Cosines to Find a Missing Side (SAS)

The most common application for finding a missing side is the SAS (Side‑Angle‑Side) case: you know two sides and the angle between them. Here is a systematic process:

  1. Identify the known sides and included angle. Label triangle vertices A, B, C with opposite sides a, b, c. Suppose you know sides a and b and the included angle C. Then the unknown side is c.
  2. Write down the appropriate formula. Use c² = a² + b² – 2ab cos(C).
  3. Plug in the known values. Substitute numbers for a, b, and angle C. Make sure your calculator is in the correct mode (degrees or radians) matching your angle measurement.
  4. Calculate step‑by‑step. Compute squares, multiply terms, then subtract. Finally take the square root to get the side length.
  5. Round appropriately. Use the required precision (typically two or three decimal places) and include units if given.

Example 1: Acute Triangle (Angle < 90°)

Given: Triangle with sides a = 8 units, b = 6 units, and included angle C = 60°. Find side c.

Solution:

  • c² = 8² + 6² – 2 × 8 × 6 × cos(60°)
  • 8² = 64, 6² = 36
  • 2 × 8 × 6 = 96, and cos(60°) = 0.5, so 96 × 0.5 = 48
  • c² = 64 + 36 – 48 = 52
  • c = √52 ≈ 7.211 (rounded to three decimal places)

So side c is approximately 7.21 units.

Example 2: Obtuse Triangle (Angle > 90°)

The Law of Cosines works equally for obtuse angles. Remember that the cosine of an obtuse angle is negative, which can affect the sign of the correction term.

Given: Triangle with sides a = 10 cm, b = 7 cm, and included angle C = 120°. Find side c.

Solution:

  • c² = 10² + 7² – 2 × 10 × 7 × cos(120°)
  • 100 + 49 = 149
  • 2 × 10 × 7 = 140, cos(120°) = –0.5, so 140 × (–0.5) = –70
  • c² = 149 – (–70) = 149 + 70 = 219
  • c = √219 ≈ 14.799 cm

Because the included angle is obtuse, the sum of squares is increased (since we subtract a negative), making the opposite side larger than it would be in a right triangle. This makes intuitive sense: spreading the two sides apart lengthens the third side.

Example 3: Real‑World Application (Finding a Distance)

Scenario: A surveyor needs to measure the distance across a lake. She can walk along two sides of the lake: 200 m east, then 150 m at an angle of 75° from the east direction (measured from her starting point). She wants the straight‑line distance from start to finish.

Solution:

Let side a = 200 m, side b = 150 m, and the included angle C = 75°. The unknown side c is the distance across the lake.

  • c² = 200² + 150² – 2 × 200 × 150 × cos(75°)
  • 40000 + 22500 = 62500
  • 2 × 200 × 150 = 60000, cos(75°) ≈ 0.258819, product ≈ 15529.14
  • c² = 62500 – 15529.14 = 46970.86
  • c = √46970.86 ≈ 216.73 m

The straight‑line distance across the lake is approximately 217 m.

If you know all three sides and need to find a missing side—well, you already have all sides. But sometimes you need to find a side that is part of a larger figure, like a quadrilateral, where one triangle’s side is unknown but linked to another triangle. In such cases, you might first solve an angle using the Law of Cosines from the SSS information, then use that angle in another SAS calculation. More commonly, the SSS case is used to find angles, not sides. However, it is worthwhile to note the relationship.

To review: given three sides a, b, c, you can find any angle by rearranging the formula:

cos(C) = (a² + b² – c²) / (2ab)

Once you have an angle, you can then apply the Law of Cosines in the SAS form for another triangle. This technique is often used in vector addition problems or navigation.

Common Mistakes and How to Avoid Them

  • Forgetting calculator mode: Most triangle problems use degrees. If your calculator is in radians, you will get wrong results. Check the mode before pressing cosine.
  • Misidentifying the included angle: The angle used in the formula must be the angle between the two known sides. If you use the opposite angle, the formula fails.
  • Sign errors with obtuse angles: Cosine of an obtuse angle is negative. If you incorrectly treat it as positive, you will get a smaller c² value (sometimes even negative) and an incorrect side.
  • Forgetting to take the square root: The Law of Cosines gives the square of the side. Always take the square root at the end.
  • Rounding too early: Keep several decimal places during intermediate steps. Round only the final answer to avoid accumulation of error.

The Law of Cosines and the Pythagorean Theorem

As mentioned, the Pythagorean theorem is a special case of the Law of Cosines when the angle is 90°. This is a beautiful example of mathematical generalization. When teaching the law, many instructors start by pointing out that the term –2ab cos(C) “corrects” the Pythagorean sum for non‑right angles. If cos(C) is positive (angle < 90°), the correction is negative, making c smaller than the hypotenuse of a right triangle with legs a and b. If cos(C) is negative (angle > 90°), the correction becomes positive, making c larger than the “right‑triangle hypotenuse.” This insight can help you estimate whether your answer makes sense.

When to Use the Law of Cosines vs. the Law of Sines

Both laws solve oblique triangles, but each has strengths.

  • Use the Law of Sines when you know two angles and any side (AAS or ASA), or two sides and a non‑included angle (SSA). However, the SSA case can lead to ambiguous results (zero, one, or two solutions).
  • Use the Law of Cosines when you know two sides and the included angle (SAS) or when you know all three sides (SSS). It is unambiguous and always gives a unique solution.

Often, you will combine both laws to fully solve a triangle: first find one side using the Law of Cosines (SAS), then find angles using either Law of Cosines again or the Law of Sines.

Practical Advice for Students and Professionals

  • Draw and label the triangle. Visualizing which sides and angles are known helps prevent misapplication.
  • Use consistent notation. Label vertices with uppercase letters A, B, C and opposite sides with lowercase a, b, c. This makes it easier to select the correct formula.
  • Check your answer with a rough estimate. For example, if the included angle is small (say 10°), the missing side should be only slightly less than the sum of the two known sides (by the triangle inequality). If your computed side is much smaller, you likely made a mistake.
  • Practice with diverse triangles: sharp acute angles, obtuse angles close to 180°, and even angles exactly 90° (to double‑check with Pythagoras).

External Resources for Further Study

These sites provide additional practice problems, step‑by‑step walkthroughs, and context for using the Law of Cosines in engineering, physics, and computer graphics.

Conclusion

The Law of Cosines is an indispensable tool for solving oblique triangles when we know two sides and the included angle (SAS) or all three sides (SSS). By extending the Pythagorean theorem to any triangle, it allows us to find unknown sides accurately—even when dealing with obtuse angles. With careful attention to calculator mode, sign conventions, and proper labeling, anyone can master this formula.

From surveying and navigation to architecture and three‑dimensional modeling, the Law of Cosines has countless applications. The more you practice with varied triangle configurations, the more intuitive it becomes. Use the external resources linked above to reinforce your understanding and test your skills with additional exercises.