Mastering a trigonometry table is a key step for GCSE maths success, helping you quickly recall standard angles and ratios. This structured approach turns complex values into easy patterns you can use in exams.
Follow the steps below, practise with the included worksheet style examples, and use the summary table to check your understanding at a glance.
| Angle (degrees) | sin | cos | tan | Memory cue |
|---|---|---|---|---|
| 0 | 0 | 1 | 0 | Line along x-axis, no height |
| 30 | 1/2 | √3/2 | 1/√3 | Hand side pattern: 1, √2, √3 over 2 |
| 45 | √2/2 | √2/2 | 1 | Isosceles right triangle ratios |
| 60 | √3/2 | 1/2 | √3 | Hand side pattern reversed from 30° |
| 90 | 1 | 0 | undefined | Vertical line, cos equals 0 |
Understanding Sine Cosine Tangent Basics
Start by learning the definitions of sine, cosine, and tangent in terms of right angled triangles. Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent.
For GCSE, you focus on special angles such as 30°, 45°, and 60°, whose ratios can be derived from exact fractions and square roots. Building fluency with these exact forms reduces reliance on calculators.
Step by Step Construction of the Table
Follow these steps to build your own trigonometry table and remember the exact values under test conditions.
Step 1 Set Up the Angle Column
List the key angles 0°, 30°, 45°, 60°, and 90° in the first column. These represent the standard angles used in most GCSE questions.
Step 2 Apply the Hand Side Pattern for Sin
Write the pattern 1, √2, √3 over 2 across the sine row. For 0° this gives 1/2 under the radical convention, which simplifies to 0, then 30° gives 1/2, 45° gives √2/2, 60° gives √3/2, and 90° gives 1.
Step 3 Derive Cos by Reversing the Pattern
Read the sine values in reverse order to fill cosine, so cos 0° = 1, cos 30° = √3/2, cos 45° = √2/2, cos 60° = 1/2, and cos 90° = 0.
Step 4 Calculate Tangent as Sin over Cos
Divide each sine value by the corresponding cosine value to obtain tangent. This gives tan 0° = 0, tan 45° = 1, and tan 60° = √3, while tan 90° is undefined due to division by zero.
Using the Table in Problem Solving
Once you can recite the table, practise translating worded questions into the correct ratio. Identify the angle, select the right fraction of sin, cos, or tan, and then compute the missing side or angle.
When solving triangle problems, label the hypotenuse, opposite, and adjacent sides clearly before substituting into the formula. This habit prevents sign errors and supports clear working in exams.
Worksheet Style Examples
Work through these short examples to see the trigonometry table gcse maths steps in action.
Example 1 Find the Opposite Side
In a right angled triangle, the angle is 30° and the hypotenuse is 10 cm. Using sin 30° = 1/2, calculate the opposite side.
Opposite = 10 × 1/2 = 5 cm.
Example 2 Find the Adjacent Side
In a right angled triangle, the angle is 45° and the hypotenuse is 8 cm. Using cos 45° = √2/2, calculate the adjacent side.
Adjacent = 8 × √2/2 = 4√2 cm.
Key Takeaways for Exam Success
- Memorise the exact values for sin, cos, and tan at 0°, 30°, 45°, 60°, and 90°.
- Use the hand side pattern to reconstruct the sine row and reverse it for cosine.
- Derive tangent by dividing sine by cosine, and note where it is undefined.
- Practise translating worded problems into the correct trigonometric ratio.
- Check your working by estimating the size of the answer using known benchmarks.
FAQ
Reader questions
How do I remember which sin value goes with which angle?
Use the hand side pattern: for sin, count fingers from 0 to 4, take the square root of that number over 2, and you get the exact values in order.
What do I do if a question asks for tan 60° as a decimal in a calculator exam?
First use the exact value √3, then switch to a calculator only if the question specifically asks for a decimal approximation to a given number of decimal places.
Why is cos 90° equal to 0 in the table?
At 90° the adjacent side shrinks to zero, so the ratio of adjacent over hypotenuse is 0, which matches the table entry and the graph of the cosine function.
Can I use the table for angles beyond 90° in GCSE exams?
For basic GCSE questions you mainly use 0° to 90°, but you can extend the table using symmetry and reference angles when required by the higher tier specifications.