Law of Sines and Cosines

Basic trigonometry (SOH CAH TOA) is perfect for right triangles, but what happens when a triangle doesn't have a $90^\circ$ angle? To solve non-right (oblique) triangles, we use the Law of Sines and the Law of Cosines. These laws allow you to find missing sides and angles in any triangle.

The Law of Sines

The Law of Sines states that the ratio of a side's length to the sine of its opposite angle is constant for all three sides of a triangle.

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

When to use it: You use the Law of Sines when you know two angles and one side (AAS or ASA), or two sides and a non-included angle (SSA).

Example: In $\triangle ABC$, $a = 8$, $\angle A = 30^\circ$, and $\angle B = 45^\circ$. Find $b$.

Solution: Set up the ratio using the Law of Sines: $\frac{8}{\sin 30^\circ} = \frac{b}{\sin 45^\circ}$

Multiply both sides by $\sin 45^\circ$ to isolate $b$: $b = \frac{8 \sin 45^\circ}{\sin 30^\circ}$

Substitute the known sine values: $b = \frac{8 \left(\frac{\sqrt{2}}{2}\right)}{\frac{1}{2}} = 8\sqrt{2}$

The Law of Cosines

The Law of Cosines generalizes the Pythagorean theorem for all triangles. It relates all three sides of a triangle to the cosine of one of its angles.

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

You can rewrite this formula for any angle:

• $a^2 = b^2 + c^2 - 2bc \cos A$
• $b^2 = a^2 + c^2 - 2ac \cos B$

When to use it: You use the Law of Cosines when you know two sides and the included angle (SAS), or when you know all three sides (SSS).

Example: In $\triangle ABC$, $a = 7$, $b = 10$, and $\angle C = 40^\circ$. Find $c$.

Solution: Since we know two sides and the included angle, we use the Law of Cosines: $c^2 = 7^2 + 10^2 - 2(7)(10) \cos 40^\circ$ $c^2 = 49 + 100 - 140(0.7660)$ $c^2 = 149 - 107.24 = 41.76$ $c \approx \sqrt{41.76} \approx 6.46$

Area of a Triangle Using Sine

You can also use trigonometry to find the area of any triangle if you know two sides and the included angle (SAS). The formula is:

$\text{Area} = \frac{1}{2}ab \sin C$

This formula works for any combination of sides and their included angle (e.g., $\frac{1}{2}bc \sin A$ or $\frac{1}{2}ac \sin B$). It is incredibly useful because you don't need to know the height of the triangle!