# Section 3.1 Law of Sines

Section Objectives

1. Use the Law of Sines to solve triangles.
2. Find the area of a triangle.
3. Use the Law of Sines in applications.

### Law of Sines

Suppose $ABC$ is a triangle with sides of lengths $a$, $b$, and $c$ that are opposite their respective vertices $A$, $B$, and $C$. Let $\alpha$, $\beta$, and $\gamma$ be the angles at $A$, $B$, and $C$. The Law of Sines says that

$\displaystyle \frac{\sin \alpha}{a} = \frac{\sin \beta}{b} = \frac{\sin \gamma}{c}$.

When the appropriate information is known, the Law of Sines can be used to solve triangles.

#### Examples

• $\gamma=102^{\circ}$, $\beta=29^{\circ}$, and $b=28$ feet. (This is an AAS case.) Determine the remaining sides and angles.

• A pole tilts toward the sun at an $8^{\circ}$ angle away from the vertical, and it casts a 22-ft shadow. The angle of elevation from the tip of the shadow to the top of the pole is $43^{\circ}$. How long is the pole? (This is an ASA case.)

### The Ambiguous Case (SSA or ASS)

A triangle is not necessarily uniquely determined in the SSA (ASS) case. There are three possibilities:

• No solution
• One solution
• Two solutions

When solving triangles, always keep in mind that the longer sides lie opposite the larger angles.

#### Examples

• $a=22$ inches, $b=12$ inches, and $\alpha =42^{\circ}$

• $a=15$ inches, $b=25$ inches, and $\alpha =85^{\circ}$

• $a=12$ meters, $b=31$ meters, and $\alpha =60^{\circ}$

### Area Formulas

It is pretty easy to derive these formulas for the area of $\triangle ABC$:

Area $= \displaystyle \frac{1}{2} bc \sin \alpha = \frac{1}{2} ab \sin \gamma = \frac{1}{2} ac \sin \beta$.

#### Examples

• Find the area of a triangular lot with two sides of lengths 90 meters and 52 meters and an included angle of $102^{\circ}$.

• On a small lake, you swim from point A to point B at a bearing of N $28^{\circ}$ E, then to point C at a bearing of N $58^{\circ}$ W, and finally due south 800 meters to return to point A\$. How far did you swim?