•

Updated Mar 22, 2026

•

7 pages

Understanding Key Trigonometric Ratios: Sine, Cosine, and Tangent

Ever wondered how builders work out roof angles or how... Show more

1 / 7

Introduction to Trigonometric Ratios

Trigonometry (or just 'trig') is everywhere around us. Engineers use it to design bridges, architects calculate roof slopes, and game developers work out character movements. The brilliant thing is, it all starts with simple right-angled triangles.

Before jumping into calculations, you've got to nail the labelling. Everything depends on which angle you're focusing on - we call this angle theta (written as θ). Get this wrong and everything else falls apart!

Quick Tip: Always identify your angle first, then label everything else relative to that angle.

The key is understanding that trigonometry only works with right-angled triangles - those with a perfect 90° corner.

Labelling Triangle Sides

Here's where students often trip up, but it's actually dead simple once you get it. You need to identify three sides relative to your chosen angle θ.

The Hypotenuse (H) is always the longest side - it's opposite the right angle and never changes. Easy to spot because it's the diagonal one.

The Opposite (O) side sits directly across from your angle θ. This one changes if you switch to looking at a different angle in the triangle.

The Adjacent (A) side is next to your angle θ (but it's not the hypotenuse). Like the opposite, this changes depending on which angle you're examining.

Remember: Opposite and Adjacent sides are always relative to your chosen angle. Switch angles, and they swap places!

The Three Main Trig Ratios

This is the heart of trigonometry - three simple ratios that connect angles to side lengths. The magic is that for any given angle, these ratios stay constant no matter how big or small your triangle is.

SOH CAH TOA is your best mate here - memorise it! It stands for:

SOH: Sine = Opposite ÷ Hypotenuse
CAH: Cosine = Adjacent ÷ Hypotenuse
TOA: Tangent = Opposite ÷ Adjacent

These trigonometric ratios are the foundation of everything. Sine connects opposite and hypotenuse, cosine links adjacent and hypotenuse, whilst tangent relates opposite and adjacent.

Exam Tip: Write "SOH CAH TOA" at the top of your exam paper - it'll save you time and stress during questions!

Working with Given Triangles

Let's see SOH CAH TOA in action with a triangle that has sides of 5, 12, and 13, focusing on angle A.

First, identify your angle - we want angle A, so θ = A. Then label the sides: hypotenuse is 13 (longest side), opposite to A is 5, and adjacent to A is 12.

Now apply the ratios:

sin(A) = 5/13 (opposite over hypotenuse)
cos(A) = 12/13 (adjacent over hypotenuse)
tan(A) = 5/12 (opposite over adjacent)

The brilliant thing is that these ratios would be exactly the same for any right-angled triangle with a matching angle, regardless of size.

Watch Out: If the question asked for angle B instead, your opposite and adjacent would swap, but the hypotenuse stays the same!

Finding Missing Side Lengths

Now for the really useful stuff - finding unknown sides using trigonometry. Say you've got a triangle with a 35° angle, hypotenuse of 15 cm, and you need to find the opposite side.

Start by identifying what you know: angle = 35°, hypotenuse = 15 cm, opposite = x (unknown). You don't need the adjacent for this problem.

Choose your ratio from SOH CAH TOA. You've got opposite and hypotenuse, so that's SOH - you need sine.

Set up your equation: sin(35°) = x/15. To find x, multiply both sides by 15: x = 15 × sin(35°).

Calculator Alert: Make sure your calculator is in DEG (degrees) mode, not RAD or GRAD - this catches loads of students out!

Solving and Key Points

Finishing the calculation: sin(35°) ≈ 0.57357, so x = 15 × 0.57357 ≈ 8.6 cm (to one decimal place).

Critical reminders that'll save your grades: SOH CAH TOA only works for right-angled triangles - no exceptions! Always check your calculator is in degrees mode before starting.

Labelling is everything - get your H, O, and A wrong and your whole answer goes wrong. The hypotenuse is always the longest side, which means sin and cos values are always less than 1.

Your problem-solving steps: label sides based on your angle, choose the right ratio, substitute values, solve for the unknown, and double-check that calculator mode!

Quick Check: If your sin or cos answer is greater than 1, something's gone wrong - probably your calculator mode or labelling!

We thought you’d never ask...

What is the Knowunity AI companion?

Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.

Where can I download the Knowunity app?

You can download the app from Google Play Store and Apple App Store.

Is Knowunity really free of charge?

That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.

Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.6/5

App Store

4.7/5

Google Play

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

Best app on earth! no words because it’s too good

Thomas R

iOS user

Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.

Basil

Android user

This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.

Rohan U

Android user

I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.

Xander S

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now

Paul T

iOS user

Stefan S

iOS user

Samantha Klich

Android user

Anna

iOS user

Best app on earth! no words because it’s too good

Thomas R

iOS user

Basil

Android user

David K

iOS user

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

Rohan U

Android user

Xander S

iOS user

Elisha

iOS user

Paul T

iOS user

Mathematics

•

Updated Mar 22, 2026

•

7 pages

Understanding Key Trigonometric Ratios: Sine, Cosine, and Tangent

Ever wondered how builders work out roof angles or how video games calculate distances? That's all trigonometry! It's basically about understanding the relationships between angles and side lengths in right-angled triangles.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

Introduction to Trigonometric Ratios

Quick Tip: Always identify your angle first, then label everything else relative to that angle.

The key is understanding that trigonometry only works with right-angled triangles - those with a perfect 90° corner.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

Labelling Triangle Sides

Here's where students often trip up, but it's actually dead simple once you get it. You need to identify three sides relative to your chosen angle θ.

The Hypotenuse (H) is always the longest side - it's opposite the right angle and never changes. Easy to spot because it's the diagonal one.

The Opposite (O) side sits directly across from your angle θ. This one changes if you switch to looking at a different angle in the triangle.

The Adjacent (A) side is next to your angle θ (but it's not the hypotenuse). Like the opposite, this changes depending on which angle you're examining.

Remember: Opposite and Adjacent sides are always relative to your chosen angle. Switch angles, and they swap places!

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

The Three Main Trig Ratios

SOH CAH TOA is your best mate here - memorise it! It stands for:

SOH: Sine = Opposite ÷ Hypotenuse
CAH: Cosine = Adjacent ÷ Hypotenuse
TOA: Tangent = Opposite ÷ Adjacent

These trigonometric ratios are the foundation of everything. Sine connects opposite and hypotenuse, cosine links adjacent and hypotenuse, whilst tangent relates opposite and adjacent.

Exam Tip: Write "SOH CAH TOA" at the top of your exam paper - it'll save you time and stress during questions!

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

Working with Given Triangles

Let's see SOH CAH TOA in action with a triangle that has sides of 5, 12, and 13, focusing on angle A.

First, identify your angle - we want angle A, so θ = A. Then label the sides: hypotenuse is 13 (longest side), opposite to A is 5, and adjacent to A is 12.

Now apply the ratios:

sin(A) = 5/13 (opposite over hypotenuse)
cos(A) = 12/13 (adjacent over hypotenuse)
tan(A) = 5/12 (opposite over adjacent)

The brilliant thing is that these ratios would be exactly the same for any right-angled triangle with a matching angle, regardless of size.

Watch Out: If the question asked for angle B instead, your opposite and adjacent would swap, but the hypotenuse stays the same!

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

Finding Missing Side Lengths

Now for the really useful stuff - finding unknown sides using trigonometry. Say you've got a triangle with a 35° angle, hypotenuse of 15 cm, and you need to find the opposite side.

Start by identifying what you know: angle = 35°, hypotenuse = 15 cm, opposite = x (unknown). You don't need the adjacent for this problem.

Choose your ratio from SOH CAH TOA. You've got opposite and hypotenuse, so that's SOH - you need sine.

Set up your equation: sin(35°) = x/15. To find x, multiply both sides by 15: x = 15 × sin(35°).

Calculator Alert: Make sure your calculator is in DEG (degrees) mode, not RAD or GRAD - this catches loads of students out!

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

Solving and Key Points

Finishing the calculation: sin(35°) ≈ 0.57357, so x = 15 × 0.57357 ≈ 8.6 cm (to one decimal place).

Critical reminders that'll save your grades: SOH CAH TOA only works for right-angled triangles - no exceptions! Always check your calculator is in degrees mode before starting.

Labelling is everything - get your H, O, and A wrong and your whole answer goes wrong. The hypotenuse is always the longest side, which means sin and cos values are always less than 1.

Your problem-solving steps: label sides based on your angle, choose the right ratio, substitute values, solve for the unknown, and double-check that calculator mode!

Quick Check: If your sin or cos answer is greater than 1, something's gone wrong - probably your calculator mode or labelling!

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students