law of cosines

Basic Concepts of Triangles

• Included Angle: - The angle formed by two sides of a triangle is called the included angle. This means it is the angle between the two given sides.

  • For example, if you have sides 'a' and 'b', the included angle would be angle 'C' (opposite side 'c').

  • To determine it, one must identify the two given sides and the specific vertex where they meet. It is crucial for applying the Law of Cosines.

Law of Cosines

• The Law of Cosines is essential for solving triangles when two sides and the included angle are known (SAS - Side-Angle-Side) or when all three sides are known (SSS - Side-Side-Side).

• It provides a direct relationship between the lengths of the sides of a triangle and the cosine of one of its angles.

• Formulas:

  • To find a side: $c^2 = a^2 + b^2 - 2ab \cos(C)$

  • To find an angle: $\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}$ (and similar permutations for angles A and B).

• Scenarios for Usage: - At least two sides and the included angle are provided (SAS). This means you know the lengths of two sides and the measure of the angle between them.

  • All three sides are provided (SSS). In this case, you can use the Law of Cosines to find any of the angles.

  • Must identify which specific information is available before applying the law to choose the correct formula permutation. The Law of Cosines is particularly useful because it does not suffer from the ambiguity problems that can arise with the Law of Sines in certain situations.

Problem Solving Approach

• Test Preparation: - On tests, students are not guided on which formula to apply, requiring them to understand how to assess the given information accurately. This involves a critical analysis of the triangle's knowns (sides and angles).

  • Students must assess what information is provided in a triangle (e.g., ASA, AAS, SAS, SSS, SSA) to determine the most appropriate solving process (Law of Sines or Law of Cosines).

  • A systematic approach includes sketching the triangle, labeling known values, and identifying the unknown quantity to be found.

Understanding Side Relationships

• When working with triangles, especially with the Law of Cosines, you may need to calculate using the square root to find side lengths.

• The equation often contains terms like $x^2$ and $a^2$ representing the square of a side length. For instance, in $c^2 = a^2 + b^2 - 2ab \cos(C)$, you calculate $c^2$ first, then take the square root to find the actual length of side $c$.

• Focus on the precise calculation method, ensuring all terms are correctly substituted and orders of operations are followed. Verify through calculator use, being mindful of correct parenthesis and function inputs (e.g., degree mode for angles).

• Remember that side lengths are always positive, so only the positive square root is considered.

Steps in Solving Triangles

• Calculating Angles: - When determining angles, students might be asked to solve for angles B or C (or A) based on given sides and relationships derived from the Law of Cosines or Law of Sines.

  • If, for example, you've used the Law of Cosines to find a side, you might then use it again to find an angle, or switch to the Law of Sines for subsequent angle calculations.

  • To find an angle using the Law of Cosines, you rearrange the formula to $\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}$, then use the inverse cosine function ($\arccos$ or $\cos^{-1}$) to find the angle A.

  • If angle ratios have been established using the Law of Sines (e.g., $\frac{a}{\sin A} = \frac{b}{\sin B}$), it may provide a straightforward approach for finding remaining angles once one angle and its opposite side are known.

Law of Sines vs. Law of Cosines

• Preference for Law of Sines:

  • Many students lean towards the Law of Sines for its perceived ease in setting up proportions and ratios ($\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$). It often feels more straightforward for finding angles and sides when certain conditions (ASA, AAS, or sometimes SSA) are met.

  • Recommendation to use it wisely to simplify problem-solving, especially when one side and two angles (ASA or AAS) are given, or after an initial application of the Law of Cosines. Its direct proportional relationship often requires fewer complex calculations compared to the Law of Cosines for subsequent steps.

  • However, awareness of the strengths and weaknesses of both the Law of Cosines and the Law of Sines is essential to choose the most efficient and unambiguous method.

• Caution with Ambiguity:

  • If using the Law of Cosines (for SAS or SSS cases), then transitioning to the Law of Sines for subsequent angle calculations may introduce ambiguity, particularly in the SSA (Side-Side-Angle) case. This is also known as the "ambiguous case" of the Law of Sines.

  • This ambiguity typically arises because the sine function gives the same positive value for both an acute angle ($\theta$) and its obtuse supplement ($180^\circ - \theta$). For example, $\sin(30^\circ) = \sin(150^\circ) = 0.5$.

  • When you use the Law of Sines to find an angle, if the value of sine is positive, your calculator will always give you the acute angle. You then have to consider if an obtuse angle is also a valid solution, which leads to two possible triangles (or no triangle). This impacts comprehension and the uniqueness of the solution.

Handling Ambiguity

• To minimize ambiguity while solving angles, especially after an initial Law of Cosines calculation: - Always solve for the largest angle first when transitioning methods (e.g., from Law of Cosines to Law of Sines).

  • The largest angle is directly opposite the largest side of the triangle.

  • Why this works: If a triangle has an obtuse angle (greater than $90^\circ$), it must be the largest angle. If you solve for the largest angle first using the Law of Sines and it's acute, then all other angles must also be acute, preventing the ambiguous case from yielding a false obtuse solution. More reliably, using the Law of Cosines to find the largest angle directly avoids this ambiguity because the inverse cosine function ($\arccos$) will give you the correct angle, whether acute or obtuse, since $\cos(\theta)$ is positive for acute angles and negative for obtuse angles, providing a unique output.

  • Solving the largest angle first cuts down the possibility of confusion and ensures clarity in the triangle's configuration by confirming whether an obtuse angle exists before proceeding with other calculations.

Conclusion of Triangle Solving Sequence

• Angle Order: - After determining the largest angle (preferably using the Law of Cosines to avoid ambiguity or carefully considering ambiguity if using the Law of Sines for this step), choose the next angle to solve.

  • A recommended sequence is to solve for the largest angle, then the second largest angle, and finally the smallest angle (or derive the smallest angle from $180^\circ - \text{Angle 1} - \text{Angle 2}$).

  • This strategy, especially prioritizing the largest angle, suggests students solve for angle C first if it's opposite the largest side, when opting to use the Law of Sines post the Law of Cosines, to mitigate the ambiguous case.

  • This sequence creates a logical flow in triangle resolution, emphasizing clarity, organization, and accuracy in the solution process, making it less prone to errors stemming from the ambiguous case.