Trigonometry: Law of Cosines

The Pythagorean theorem is a powerful tool, but it only works for one specific type of triangle: the right triangle. The real world is full of oblique triangles—triangles without a right angle. To solve these, you need a more powerful, generalized tool. The Law of Cosines is that tool, allowing you to find unknown sides and angles in any triangle, whether it's acute, obtuse, or right. Mastering it is essential for fields from surveying and engineering to physics and computer graphics, where triangular relationships are foundational but rarely perfect.

From Pythagoras to a General Rule

The Law of Cosines is a direct extension of the Pythagorean theorem. Recall that for a right triangle with hypotenuse $c$, the theorem states $c^2=a^2+b^2$. The Law of Cosines modifies this formula to account for any angle $C$ opposite side $c$. For any triangle with sides $a$, $b$, and $c$ opposite angles $A$, $B$, and $C$ respectively, the formula is:

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

This elegant equation says: the square of one side equals the sum of the squares of the other two sides, minus twice the product of those two sides and the cosine of the angle between them. The critical insight is the term $-2ab\cos (C)$. When angle $C=90^{\circ }$, $\cos (90^{\circ })=0$, and the entire term vanishes, leaving you with the classic Pythagorean theorem. Therefore, the Law of Cosines generalizes the Pythagorean theorem to all triangles.

It’s vital to correctly label your triangle. The side you are solving for (or the side opposite the angle you know) is isolated on the left. The other two sides and the cosine of their included angle are on the right.

Solving SAS Triangles: Finding the Missing Side

The most straightforward application is solving an SAS (Side-Angle-Side) triangle. You are given two sides and the measure of the angle between them. Your goal is to find the third side directly opposite the known angle.

Example: A triangle has sides $a=7$ units and $b=10$ units, with the included angle $C=34^{\circ }$. Find side $c$.

1. Identify the knowns: We have sides $a$ and $b$ and the included angle $C$. We want side $c$, opposite angle $C$.
2. Apply the formula: $$c^2=a^2+b^2-2ab\cos (C)$$
3. Substitute values: $$c^2=7^2+10^2-2(7)(10)\cos (34^{\circ })$$
4. Calculate: $$c^2=49+100-140\cos (34^{\circ })$$ $$c^2\approx 149-140(0.8290)$$ $$c^2\approx 149-116.06=32.94$$
5. Solve for c: $$c\approx \sqrt{32.94}\approx 5.74\text{ units}$$

This process is direct and unambiguous. Once you have the third side, you could then use the Law of Sines or Cosines again to find the remaining angles.

Solving SSS Triangles: Finding an Angle

The Law of Cosines is also the primary method for solving SSS (Side-Side-Side) triangles, where you know all three sides but none of the angles. Here, you rearrange the formula to solve for the cosine of an angle.

The rearranged formula is: $$\cos (C)=\frac{a^2+b^2-c^2}{2ab}$$

Example: A triangle has sides $a=8$, $b=14$, and $c=17$. Find angle $C$.

1. Identify the setup: Side $c=17$ is opposite the angle we want, $C$. Sides $a$ and $b$ are the other two.
2. Apply the rearranged formula: $$\cos (C)=\frac{8^2+14^2-17^2}{2(8)(14)}$$
3. Calculate: $$\cos (C)=\frac{64+196-289}{224}$$ $$\cos (C)=\frac{-29}{224}\approx -0.1295$$
4. Find the angle: Use the inverse cosine function. $C={\cos }^{-1}(-0.1295)\approx 97.4^{\circ }$.

The negative cosine tells you immediately that angle $C$ is obtuse (greater than $90^{\circ }$). This is a quick way to determine triangle type from sides alone. In any triangle with sides $a$, $b$, and $c$ (where $c$ is the longest side):

• If $c^2<a^2+b^2$, the triangle is acute (all angles < $90^{\circ }$).
• If $c^2=a^2+b^2$, the triangle is right (Pythagorean theorem holds).
• If $c^2>a^2+b^2$, the triangle is obtuse (the angle opposite the longest side is > $90^{\circ }$).

Practical Applications and the Ambiguous Case

Beyond textbook problems, the Law of Cosines is used in navigation to find the distance between two points given bearings from a third, in mechanical design to calculate forces in non-right-angled trusses, and in computer vision to determine object orientation.

A crucial strategic point involves the ambiguous case of the Law of Sines. When you are given SSA (two sides and a non-included angle), the Law of Sines can yield two possible solutions, one acute and one obtuse. The Law of Cosines can be used to resolve this ambiguity efficiently. While solving directly with the Law of Cosines for SSA requires solving a quadratic equation, you can use it as a check: after finding a potential side or angle with the Law of Sines, use the Law of Cosines to verify the consistency of all three sides and angles. More directly, if you find yourself with SSA, you can often avoid ambiguity by using the Law of Cosines from the outset to find the side opposite the given angle, though it requires more algebraic steps.

Common Pitfalls

1. Mislabeling the Triangle: The most frequent error is misassigning sides to angles. Remember, side $a$ is opposite angle $A$, side $b$ is opposite angle $B$, and side $c$ is opposite angle $C$. The angle in the formula must be the one opposite the side you have on the left of the equation. Double-check your labeling before you substitute.

1. Using the Wrong Formula for the Given Information: The Law of Sines is useful for AAS and ASA cases and is simpler for those. A common mistake is forcing the Law of Cosines when the Law of Sines is more efficient. Use Law of Cosines for SAS and SSS scenarios. For SSA, be aware of the potential for two triangles and use the Law of Cosines to verify your single solution from the Law of Sines, or to solve the quadratic that arises.

1. Calculator in the Wrong Mode: Angles in trigonometry problems can be measured in degrees or radians. If your angle is given in degrees (like $34^{\circ }$), your calculator must be in Degree mode. If you compute $\cos (34)$ in Radian mode, you'll get an incorrect value, leading to a wrong answer. Always verify your calculator's angle setting.

1. Algebraic Errors with the Rearranged Formula: When solving for an angle using $\cos (C)=(a^2+b^2-c^2)/(2ab)$, ensure the correct side is subtracted. The side opposite the angle you seek ($c$) is the one being subtracted from the sum of squares of the other two sides ($a^2+b^2$). Mixing this up will flip the sign of your cosine and give you the supplementary angle.

Summary

• The Law of Cosines, $c^2=a^2+b^2-2ab\cos (C)$, is the definitive generalization of the Pythagorean theorem, working for all triangle types: acute, right, and obtuse.
• It is the primary tool for solving SAS triangles (to find the missing side) and SSS triangles (to find any angle after rearranging the formula).
• You can determine a triangle's type by comparing the square of the longest side to the sum of squares of the other two sides: $c^2<a^2+b^2$ (acute), $c^2=a^2+b^2$ (right), $c^2>a^2+b^2$ (obtuse).
• Always ensure your triangle is labeled correctly (sides opposite their corresponding angles) and your calculator is in the correct angle mode (typically degrees) to avoid fundamental errors.
• Strategically, prefer the Law of Cosines for SAS and SSS cases, and understand its role in checking or resolving the ambiguous SSA case that arises with the Law of Sines.