Law of Sines Calculator

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

This Law of Sines Calculator takes two angles and the side opposite one of them and works out the two missing sides, showing the working line by line. The third angle comes first, because the three angles of a triangle always add up to a half turn. The law then ties each side to the sine of the angle facing it, so rearranging it makes a missing side the subject. Following the same pattern every time keeps each side matched to the right angle.

Step-by-step method

  1. Set up the problem.
  2. Write the angle sum of a triangle.
  3. Subtract the two known angles to find the third.
  4. Write the Law of Sines.
  5. Rearrange the Law of Sines to make a missing side the subject.
  6. Work out the sine of each angle.
  7. Substitute the values and divide to find each missing side.

Formula:

\(\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\)
See Example 1 Hide Example 1

Problem

\(A = 35^{\circ},\; B = 65^{\circ},\; a = 8\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the angle sum of a triangle.
  3. Subtract the two known angles to find the third.
  4. Write the Law of Sines.
  5. Rearrange the Law of Sines to make a missing side the subject.
  6. Work out the sine of each angle.
  7. Substitute the values and divide to find each missing side.

Step 1

Step 1 - Set up the problem.

In this problem: Two angles and the side opposite one of them are known: \(A = 35^{\circ}\), \(B = 65^{\circ}\) and \(a = 8\), and sides \(b\) and \(c\) are missing.

\(A = 35^{\circ},\; B = 65^{\circ},\; a = 8\)

Step 2

Step 2 - Write the angle sum of a triangle.

In this problem: The three angles of any triangle add up to \(180^{\circ}\), which pins down the third angle.

\(A + B + C = 180^{\circ}\)

Step 3

Step 3 - Subtract the two known angles to find the third.

In this problem: Taking \(35^{\circ}\) and \(65^{\circ}\) away from \(180^{\circ}\) leaves \(80^{\circ}\).

\(C = 180^{\circ} - 35^{\circ} - 65^{\circ} = 80^{\circ}\)

Step 4

Step 4 - Write the Law of Sines.

In this problem: The Law of Sines says each side and the sine of the angle opposite it keep the same ratio, so \(\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\).

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

Step 5

Step 5 - Rearrange the Law of Sines to make a missing side the subject.

In this problem: Multiplying both sides of \(\frac{a}{\sin(A)} = \frac{b}{\sin(B)}\) by \(\sin(B)\) makes the missing side the subject, giving \(b = a \times \frac{\sin(B)}{\sin(A)}\).

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

Step 6

Step 6 - Work out the sine of each angle.

In this problem: The sines of the three angles are \(\sin(35^{\circ}) = 0.57\), \(\sin(65^{\circ}) = 0.91\) and \(\sin(80^{\circ}) = 0.98\).

\(\sin(A) = 0.57,\; \sin(B) = 0.91,\; \sin(C) = 0.98\)

Step 7a

Step 7a - Substitute the values and divide to find each missing side.

In this problem: For side \(b\), substituting gives \(8 \times \frac{0.91}{0.57}\), which works out to \(12.64\).

\(b = 8 \times \frac{0.91}{0.57} \approx 12.64\)

Step 7b

Step 7b - Substitute the values and divide to find each missing side.

In this problem: For side \(c\), substituting gives \(8 \times \frac{0.98}{0.57}\), which works out to \(13.74\).

\(c = 8 \times \frac{0.98}{0.57} \approx 13.74\)

Final Answer

\(b \approx 12.64,\; c \approx 13.74\)
See Example 2 Hide Example 2

Problem

\(A = 50^{\circ},\; B = 55^{\circ},\; a = 14\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the angle sum of a triangle.
  3. Subtract the two known angles to find the third.
  4. Write the Law of Sines.
  5. Rearrange the Law of Sines to make a missing side the subject.
  6. Work out the sine of each angle.
  7. Substitute the values and divide to find each missing side.

Step 1

Step 1 - Set up the problem.

In this problem: Two angles and the side opposite one of them are known: \(A = 50^{\circ}\), \(B = 55^{\circ}\) and \(a = 14\), and sides \(b\) and \(c\) are missing.

\(A = 50^{\circ},\; B = 55^{\circ},\; a = 14\)

Step 2

Step 2 - Write the angle sum of a triangle.

In this problem: The three angles of any triangle add up to \(180^{\circ}\), which pins down the third angle.

\(A + B + C = 180^{\circ}\)

Step 3

Step 3 - Subtract the two known angles to find the third.

In this problem: Taking \(50^{\circ}\) and \(55^{\circ}\) away from \(180^{\circ}\) leaves \(75^{\circ}\).

\(C = 180^{\circ} - 50^{\circ} - 55^{\circ} = 75^{\circ}\)

Step 4

Step 4 - Write the Law of Sines.

In this problem: The Law of Sines says each side and the sine of the angle opposite it keep the same ratio, so \(\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\).

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

Step 5

Step 5 - Rearrange the Law of Sines to make a missing side the subject.

In this problem: Multiplying both sides of \(\frac{a}{\sin(A)} = \frac{b}{\sin(B)}\) by \(\sin(B)\) makes the missing side the subject, giving \(b = a \times \frac{\sin(B)}{\sin(A)}\).

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

Step 6

Step 6 - Work out the sine of each angle.

In this problem: The sines of the three angles are \(\sin(50^{\circ}) = 0.77\), \(\sin(55^{\circ}) = 0.82\) and \(\sin(75^{\circ}) = 0.97\).

\(\sin(A) = 0.77,\; \sin(B) = 0.82,\; \sin(C) = 0.97\)

Step 7a

Step 7a - Substitute the values and divide to find each missing side.

In this problem: For side \(b\), substituting gives \(14 \times \frac{0.82}{0.77}\), which works out to \(14.97\).

\(b = 14 \times \frac{0.82}{0.77} \approx 14.97\)

Step 7b

Step 7b - Substitute the values and divide to find each missing side.

In this problem: For side \(c\), substituting gives \(14 \times \frac{0.97}{0.77}\), which works out to \(17.65\).

\(c = 14 \times \frac{0.97}{0.77} \approx 17.65\)

Final Answer

\(b \approx 14.97,\; c \approx 17.65\)

Example Problems

Click one to load it into the box and solve it.