Law of Cosines and Sines

Law of Cosines and Law of Sines

The Law of Cosines and Law of Sines are crucial tools in trigonometry, used to solve triangles that are not necessarily right-angled. They relate the angles and sides of a triangle, and they are especially useful for solving oblique triangles (triangles that are not right-angled).

1. Law of Cosines

The Law of Cosines relates the sides and angles of a triangle by expressing the length of a side in terms of the other two sides and the cosine of the included angle.

For any triangle with sides $a$, $b$, and $c$, and corresponding angles $A$, $B$, and $C$, the Law of Cosines states:

$$c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$$ $$b^2 = a^2 + c^2 - 2ac \cdot \cos(B)$$ $$a^2 = b^2 + c^2 - 2bc \cdot \cos(A)$$

Where:

• $a$, $b$, and $c$ are the lengths of the sides of the triangle.
• $A$, $B$, and $C$ are the angles opposite to the sides $a$, $b$, and $c$, respectively.

Uses of the Law of Cosines:

• Finding a side when two sides and the included angle are known.
• Finding an angle when all three sides are known (using the inverse cosine function).

Example 1: Finding the third side of a triangle.

Given a triangle with sides $a = 5$, $b = 7$, and the included angle $C = 60^\circ$, find side $c$.

Using the Law of Cosines formula:

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

Substitute the known values:

$$c^2 = 5^2 + 7^2 - 2(5)(7) \cdot \cos(60^\circ)$$

Since $\cos(60^\circ) = 0.5$, we have:

$$c^2 = 25 + 49 - 70 \times 0.5$$ $$c^2 = 25 + 49 - 35$$ $$c^2 = 39$$ $$c = \sqrt{39} \approx 6.24$$

So, side $c \approx 6.24$.

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2. Law of Sines

The Law of Sines relates the sides of a triangle to the sines of its angles. It is used to solve triangles when two angles and one side are known, or two sides and a non-included angle are known.

The Law of Sines states that:

$$\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$$

Where:

• $a$, $b$, and $c$ are the sides of the triangle.
• $A$, $B$, and $C$ are the angles opposite those sides.

Uses of the Law of Sines:

• Finding an angle when two sides and an angle (non-included) are known (called the SSA case).
• Finding a side when two angles and one side are known.

Example 2: Finding an angle when two sides and an opposite angle are known.

Given a triangle with sides $a = 8$, $b = 10$, and angle $A = 30^\circ$, find angle $B$.

Using the Law of Sines:

$$\frac{a}{\sin(A)} = \frac{b}{\sin(B)}$$

Substitute the known values:

$$\frac{8}{\sin(30^\circ)} = \frac{10}{\sin(B)}$$

Since $\sin(30^\circ) = 0.5$, we have:

$$\frac{8}{0.5} = \frac{10}{\sin(B)}$$ $$16 = \frac{10}{\sin(B)}$$

Solve for $\sin(B)$:

$$\sin(B) = \frac{10}{16} = 0.625$$

Now, take the inverse sine:

$$B = \sin^{-1}(0.625) \approx 38.68^\circ$$

So, angle $B \approx 38.68^\circ$.

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3. Ambiguous Case (SSA Case)

The Ambiguous Case occurs when you are given two sides and a non-included angle (SSA case). In this case, there may be:

1. One solution (a unique triangle).
2. Two solutions (two possible triangles).
3. No solution (no possible triangle).

For example, given side $a$, side $b$, and angle $A$, the Law of Sines might result in two possible values for angle $B$. Depending on the situation, this could lead to two possible triangles.

Example of the Ambiguous Case: Given $a = 8$, $b = 10$, and $A = 30^\circ$, let's find angle $B$. Using the Law of Sines:

$$\frac{8}{\sin(30^\circ)} = \frac{10}{\sin(B)}$$ $$\frac{8}{0.5} = \frac{10}{\sin(B)}$$ $$16 = \frac{10}{\sin(B)}$$ $$\sin(B) = \frac{10}{16} = 0.625$$

So, $\sin(B) = 0.625$, and taking the inverse sine:

$$B = \sin^{-1}(0.625) \approx 38.68^\circ$$

Now, since $B = 38.68^\circ$ is possible, we can also find a second solution by noting that $B' = 180^\circ - B$, which gives:

$$B' = 180^\circ - 38.68^\circ \approx 141.32^\circ$$

This means that there are two possible values for angle $B$, leading to two possible triangles.

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4. Summary

• Law of Cosines: Useful when you know two sides and the included angle (SAS) or all three sides (SSS).

  • Formula: $c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$

• Law of Sines: Useful when you know two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA).

  • Formula: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$

• Ambiguous Case (SSA): When given two sides and a non-included angle, there may be one, two, or no solutions.

Both laws are powerful tools for solving non-right triangles, and they allow us to find unknown sides or angles in a triangle using only the information about the other sides and angles.