# The Law of Sines

The law of sines is an equation relating the lengths of the sides of a triangle to the sides of its angles. In this post, we will learn more about the law of sines. So join us.

When you already know the measurements of 2 factors in a triangular as well as one of the sides, like in the case of ASA or AAS, you may apply the law of sines to determine the measures of the other two angles and the other side. This law is based on the ratio of the side of the triangle and the angles that are opposite to each other. The greater the width of a side, the greater the width of its opposite angle. The largest side is always on the other side of the most acute angle.

## Related Topics

- How to Find Missing Sides and Angles of a Right Triangle
- How to Evaluate Trigonometric Function
- How to Solve Angles and Angle Measure

## Step by step guide to the law of sines

**The law of sines is:**

In a right oblique triangle, if the measurements of the angles are \(A\), \(B\), and \(C\), and the measurements of the length of the sides opposite the corresponding angles are \(a\), \(b\), and \(c\), then the length of each side to the sine of the opposing angle must all be equal.

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

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

### The Law of Sines – Example 1:

In the ABC triangle, find the sides and the angle.

All angles in a triangle sum up to \(180\) degrees. Then:

\(30+20+x=180→ 50+x=180→x=180-50=130 ^\circ \)

To find sides use the law of sines: \(\frac {a}{sin\ A}=\frac {b}{sin\ B}=\frac {c}{sin\ C}\)

\(\frac {a}{sin 30}=\frac {45}{sin\ 20}= \frac {c}{sin 130}\)

Now, use proportional ratios: \(\frac {a}{b}=\frac{c}{d} → a×d=c×b\)

\(\frac {a}{sin 30}=\frac {45}{sin\ 20}\) → \(a=\frac {45 × sin\ 30}{sin 20}=\frac {45 × 0.5}{0.34}=\frac {22.5}{0.34}≅66.18 m\)

\(\frac {45}{sin\ 20}= \frac {c}{sin 130}\) → \(c=\frac {45 × sin 130}{sin 20}=\frac {45 × 0.77}{0.34}=\frac {34.65}{0.34}≅101.91 m\)

### The Law of Sines – Example 2:

Find the sides and the angle in the ABC triangle.

All angles in a triangle sum up to \(180\) degrees. Then:

\(75+42+x=180→ 117+x=180→x=180-117=63 ^\circ \)

To find sides use the law of sines: \(\frac {a}{sin\ A}=\frac {b}{sin\ B}=\frac {c}{sin\ C}\)

\(\frac {22}{sin\ 75}=\frac {b}{sin\ 42}= \frac {c}{sin\ 63}\)

Now, use proportional ratios: \(\frac {a}{b}=\frac{c}{d} → a×d=c×b\)

\(\frac {22}{sin\ 75}=\frac {b}{sin\ 42} → b=\frac {22 × sin\ 42 } {sin\ 75} =\frac{22 × 0.67}{0.96}=\frac {14.74}{0.96}=15.35\ cm\)

\(\frac {22}{sin\ 75}= \frac {c}{sin\ 63} → c=\frac {22 × sin\ 63 } {sin\ 75} =\frac{22 × 0.9}{0.96}=\frac {19.8}{0.96}=20.62\ cm\)

## Exercises for the Law of Sines

### Find the side of c in the ABC triangle.

1.

2.

3.

- \(\color{blue}{73.33}\)
- \(\color{blue}{6.51}\)
- \(\color{blue}{20.53}\)

### More math articles

- Best Calculator For 11th Grade Students
- Top 10 Free Websites for PSAT / NMSQT Math Preparation
- 7th Grade NHSAS Math Worksheets: FREE & Printable
- FREE 3rd Grade Common Core Math Practice Test
- A Comprehensive Collection of Free ASVAB Math Practice Tests
- Full-Length 6th Grade ACT Aspire Math Practice Test
- Overview of the PSAT / NMSQT Mathematics Test
- Top Calculators for the GED 2022: Quick Review
- How to Solve Radical Equations? (+FREE Worksheet!)
- How to Find the Volume of Spheres? (+FREE Worksheet!)

## What people say about "The Law of Sines"?

No one replied yet.