Trigonometric Ratios
Sine, cosine and tangent are ratios of the sides of a right-angled triangle linked to one of its angles.
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Play with the controls to see the idea in action.
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Sine, cosine and tangent are ratios of the sides of a right-angled triangle linked to one of its angles.
- SOH CAH TOA helps you remember them
- Sine is opposite over hypotenuse
- Use inverse functions to find angles
🧠 Quick quiz
Three questions. Score 2 or more to complete the topic.
What is Trigonometric Ratios?
Within Mathematics, which covers numbers, algebra, geometry, trigonometry and statistics from first principles to exam level, this topic explains Trigonometric Ratios. Maths builds step by step, so each topic here gives you a rule, a picture and a quick check you can use on homework and in exams.
Check that you can explain Trigonometric Ratios in your own words and give one example from everyday life. If you can do that, you have understood the core idea.
Understanding Trigonometric Ratios gives you a reliable method you will reuse in later maths units, in science lessons and in data work.
Trigonometric Ratios in detail
In a right-angled triangle, the sine, cosine and tangent ratios link an angle to the sides: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse and tan θ = opposite/adjacent, remembered as SOH CAH TOA. The sides are labelled relative to the angle you are working with.
Use the ratios to find unknown sides or, with the inverse functions sin⁻¹, cos⁻¹ and tan⁻¹, unknown angles. Make sure your calculator is in degree mode. Trigonometry is used to find heights of buildings, slopes of ramps and distances in surveying and navigation.
How the interactive helps you understand Trigonometric Ratios
Change the angle or let it animate. The horizontal and vertical lengths on the circle show the cosine and sine values for that angle.
Before you change anything, predict the result. Then change one value at a time and check whether your prediction was right. This is a quick way to build a correct mental model of Trigonometric Ratios.
Real-life uses and examples
- Use Trigonometric Ratios when comparing prices, discounts or recipe quantities in everyday life.
- Explain Trigonometric Ratios to a friend with a simple picture, such as a row of tiles or a bag of sweets.
- Spot Trigonometric Ratios in exam questions, where it often sits inside a short word problem.
Common mistakes to avoid
- Forgetting units, signs or brackets when you apply Trigonometric Ratios.
- Memorising the steps for Trigonometric Ratios without checking why they work.
- Learning the definition without testing it. Use the quiz and the interactive to check that you can apply the idea.
Frequently asked questions about Trigonometric Ratios
What is Trigonometric Ratios?
Sine, cosine and tangent are ratios of the sides of a right-angled triangle linked to one of its angles.
What are the most important points about Trigonometric Ratios?
The main points are: SOH CAH TOA helps you remember them; Sine is opposite over hypotenuse; Use inverse functions to find angles.
Why is Trigonometric Ratios useful?
Understanding Trigonometric Ratios gives you a reliable method you will reuse in later maths units, in science lessons and in data work.
How can I practise Trigonometric Ratios?
Open the interactive at the top of this page and change the controls one at a time. Predict the result before each change, then take the three-question quiz. A score of two or more marks the topic as complete on this device.
Which topics are related to Trigonometric Ratios?
Related topics in Mathematics include Area of a Triangle Using Sine, Angles of Elevation and Depression, Pyramids, Cones and Spheres.