Law of Sines Calculator

Published on: February 2, 2025
Final Answer: Free • Full Steps: Plus

This Law of Sines Calculator helps you find missing side lengths in a triangle when two angles and one opposite side are known. It uses the formula a / sin(A) = b / sin(B) = c / sin(C) to relate side lengths to their opposite angles. First find the third angle, then use the law of sines to calculate the missing sides. It is a simple way to check answers, understand the law of sines, and practise basic trigonometry step by step.

Step-by-step method

  1. Identify what is given.
  2. Write the Law of Sines.
  3. Find the third angle.
  4. Use the Law of Sines to find the missing sides.

Formula:

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

Example 1:

\(A = 40^{\circ},\; B = 60^{\circ},\; a = 10\)

Step 1 - Identify what is given.

In this problem: The given values are \(A = 40^{\circ}\), \(B = 60^{\circ}\), and \(a = 10\).

\(A = 40^{\circ},\; B = 60^{\circ},\; a = 10\)

Step 2 - Write the Law of Sines.

In this problem: Use the Law of Sines: \(\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\).

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

Step 3 - Find the third angle.

In this problem: Angles in a triangle add to \(180^{\circ}\), so \(C = 180^{\circ} - 40^{\circ} - 60^{\circ} = 80^{\circ}\).

\(C = 180^{\circ} - 40^{\circ} - 60^{\circ} = 80^{\circ}\)

Step 4 - Use the Law of Sines to find the missing sides.

In this problem: Rearrange for each side: \(b = a \times \frac{\sin(B)}{\sin(A)} = 10 \times \frac{0.87}{0.64} \approx 13.47\) and \(c = a \times \frac{\sin(C)}{\sin(A)} = 10 \times \frac{0.98}{0.64} \approx 15.32\).

\(b \approx 13.47,\; c \approx 15.32\)

Final answer:

\(b \approx 13.47,\; c \approx 15.32\)

Example 2:

\(A = 30^{\circ},\; B = 70^{\circ},\; a = 12\)

Step 1 - Identify what is given.

In this problem: The given values are \(A = 30^{\circ}\), \(B = 70^{\circ}\), and \(a = 12\).

\(A = 30^{\circ},\; B = 70^{\circ},\; a = 12\)

Step 2 - Write the Law of Sines.

In this problem: Use the Law of Sines: \(\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\).

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

Step 3 - Find the third angle.

In this problem: Angles in a triangle add to \(180^{\circ}\), so \(C = 180^{\circ} - 30^{\circ} - 70^{\circ} = 80^{\circ}\).

\(C = 180^{\circ} - 30^{\circ} - 70^{\circ} = 80^{\circ}\)

Step 4 - Use the Law of Sines to find the missing sides.

In this problem: Rearrange for each side: \(b = a \times \frac{\sin(B)}{\sin(A)} = 12 \times \frac{0.94}{0.5} \approx 22.55\) and \(c = a \times \frac{\sin(C)}{\sin(A)} = 12 \times \frac{0.98}{0.5} \approx 23.64\).

\(b \approx 22.55,\; c \approx 23.64\)

Final answer:

\(b \approx 22.55,\; c \approx 23.64\)
See Example 1 Hide Example 1

Problem

\(A = 40^{\circ},\; B = 60^{\circ},\; a = 10\)

Approach

Step-by-step method

  1. Identify what is given.
  2. Write the Law of Sines.
  3. Find the third angle.
  4. Use the Law of Sines to find the missing sides.

Step 1

Step 1 - Identify what is given.

In this problem: The given values are \(A = 40^{\circ}\), \(B = 60^{\circ}\), and \(a = 10\).

\(A = 40^{\circ},\; B = 60^{\circ},\; a = 10\)

Step 2

Step 2 - Write the Law of Sines.

In this problem: Use the Law of Sines: \(\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\).

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

Step 3

Step 3 - Find the third angle.

In this problem: Angles in a triangle add to \(180^{\circ}\), so \(C = 180^{\circ} - 40^{\circ} - 60^{\circ} = 80^{\circ}\).

\(C = 180^{\circ} - 40^{\circ} - 60^{\circ} = 80^{\circ}\)

Step 4

Step 4 - Use the Law of Sines to find the missing sides.

In this problem: Rearrange for each side: \(b = a \times \frac{\sin(B)}{\sin(A)} = 10 \times \frac{0.87}{0.64} \approx 13.47\) and \(c = a \times \frac{\sin(C)}{\sin(A)} = 10 \times \frac{0.98}{0.64} \approx 15.32\).

\(b \approx 13.47,\; c \approx 15.32\)

Final Answer

\(A = 40^{\circ},\; B = 60^{\circ},\; a = 10 = b \approx 13.47,\; c \approx 15.32\)
See Example 2 Hide Example 2

Problem

\(A = 30^{\circ},\; B = 70^{\circ},\; a = 12\)

Approach

Step-by-step method

  1. Identify what is given.
  2. Write the Law of Sines.
  3. Find the third angle.
  4. Use the Law of Sines to find the missing sides.

Step 1

Step 1 - Identify what is given.

In this problem: The given values are \(A = 30^{\circ}\), \(B = 70^{\circ}\), and \(a = 12\).

\(A = 30^{\circ},\; B = 70^{\circ},\; a = 12\)

Step 2

Step 2 - Write the Law of Sines.

In this problem: Use the Law of Sines: \(\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\).

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

Step 3

Step 3 - Find the third angle.

In this problem: Angles in a triangle add to \(180^{\circ}\), so \(C = 180^{\circ} - 30^{\circ} - 70^{\circ} = 80^{\circ}\).

\(C = 180^{\circ} - 30^{\circ} - 70^{\circ} = 80^{\circ}\)

Step 4

Step 4 - Use the Law of Sines to find the missing sides.

In this problem: Rearrange for each side: \(b = a \times \frac{\sin(B)}{\sin(A)} = 12 \times \frac{0.94}{0.5} \approx 22.55\) and \(c = a \times \frac{\sin(C)}{\sin(A)} = 12 \times \frac{0.98}{0.5} \approx 23.64\).

\(b \approx 22.55,\; c \approx 23.64\)

Final Answer

\(A = 30^{\circ},\; B = 70^{\circ},\; a = 12 = b \approx 22.55,\; c \approx 23.64\)