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# Find missing angles using the sine rule
- URL: https://www.esheets.io/sine-rule-missing-angles/
- Published: 2026-05-02T11:10:24.000Z
- Updated: 2026-07-09T20:25:51.000Z
- Author: Richard Linnington 2
- Tags: Maths, Geometry and Shape

The sine rule lets you calculate unknown angles in any triangle, making it a crucial tool when right-angled trigonometry isn't an option. This is an essential skill for fields like aviation and marine navigation, where calculating the exact turning angle or heading is critical for plotting a safe course. [Jump to the questions](#practise-now)

[Looking for questions on missing lengths with the sine rule?](https://www.esheets.io/sine-rule-missing-lengths/)

## Practise now

Worksheet preview and key skills 

### Worksheet preview

Practise the sine rule for missing angles with this self-marking maths worksheet.

The interactive worksheet below generates questions, gives instant feedback, and lets students record their score.

#### What you’ll practise

- Identifying a matching angle-side pair.
- Choosing the sine rule for missing angles.
- Substituting known values into the sine rule.
- Using inverse sine to find the angle.

Use the interactive worksheet below, or read the [Topic guide](#topic-guide) for the method and worked example.

## Topic guide

### What this worksheet practises

This worksheet provides practice on using the Sine Rule to calculate a missing angle in any triangle. Because we are looking for an angle, we flip the standard formula upside down. This makes the algebra much easier to solve.

### Key method

The flipped Sine Rule for missing angles is: **sin(A) / a = sin(B) / b**

- **Label your triangle:** Label the missing angle you want to find 'A', and the side directly opposite it 'a'. Label the known angle 'B', and the side directly opposite it 'b'.
- Write out the flipped formula and substitute your numbers in.
- To isolate 'sin(A)', move the side 'a' from the bottom left across to the top right. It becomes a multiplication.
- Your calculation will look like: **sin(A) = (sin(B) / b) × a**.
- This gives you the value of *sin(A)*. To find the actual angle 'A', you must use the inverse sine function (**sin&supmin;¹**) on your calculator.

### Worked example

**A triangle has a side of 8cm opposite a missing angle 'x'. It has another side of 12cm opposite a known angle of 50°. Calculate angle 'x'.**

Step 1: Label and substitute into the flipped formula.

sin(x) / 8 = sin(50) / 12

Step 2: Rearrange to get sin(x) on its own. Multiply both sides by 8.

sin(x) = (sin(50) / 12) × 8

Step 3: Calculate the value of sin(x).

sin(x) = 0.06383... × 8

sin(x) = 0.5106...

Step 4: Use inverse sine (shift-sin on most calculators) to find the angle.

x = sin&supmin;¹(0.5106...)

x = 30.70...

The final answer is 30.7° (to 1 d.p.).

### Common mistakes to avoid

The most catastrophic mistake is forgetting the final step. A student will correctly calculate the value of sin(x) as 0.5106 and write that down as their final answer. An angle of 0.5 degrees inside a triangle is almost impossible to draw. You must remember to use sin&supmin;¹ to turn the decimal ratio back into an actual degree measurement.

### Things to remember

If you type your calculation into the calculator and it says "Maths ERROR" or "Syntax Error", it means you have made a mistake in your rearrangement, and your value for sin(x) is greater than 1\. The sine of an angle can never be larger than 1\. Check your fractions and try again.