Mastering the Law of Sines: Explanation and Practice Problems

Chelsea

12/8/2025

Trigonometry

The Sine Law

•

Dec 8, 2025

•

5 pages

Mastering the Law of Sines: Explanation and Practice Problems

Chelsea

@zuncho

The Sine Law is a powerful tool for solving triangles... Show more

1 / 5

MBF3C: Mathematics of Personal Finance
Day 4: The Sine Law
I
I
I
I
I
I
I
THE SINE LAW
So far, we've used trigonometric ratios to solve right

Understanding the Sine Law

Ever wondered how to solve triangles that aren't right-angled? The Sine Law is your answer! This formula creates a relationship between the sides of any triangle and their opposite angles.

The Sine Law can be written two ways depending on what you need to find:

For finding sides: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$
For finding angles: $\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}$

When using the Sine Law, remember that each side pairs with its opposite angle (side a faces angle A). You only need to work with two pairs at a time, not all three.

💡 Quick Tip: Think of the Sine Law as a proportion - if you know one side-angle pair, you can find another side or angle using that relationship.

Let's see it in action! To find the unknown side x in a triangle with angles 80° and 60° and side length 7, we set up: $\frac{x}{\sin(80°)} = \frac{7}{\sin(60°)}$ . Solving for x gives us 7.96 units.

Finding Missing Angles Using the Sine Law

Now let's flip things around and use the Sine Law to find missing angles. When you know two sides and an angle opposite one of them, you can discover the missing angle.

For finding angles, we rearrange the formula to: $\frac{\sin(A)}{a} = \frac{\sin(B)}{b}$ . This version puts the angles in the numerator position, making it easier to solve for an unknown angle.

Here's how to solve a problem: Imagine a triangle with sides 8 and 10, and an angle of 75° opposite the side of length 10. To find the angle m opposite the side of length 8, we set up: $\frac{\sin(m°)}{8} = \frac{\sin(75°)}{10}$

Solving step by step:

Multiply both sides by 8: $\sin(m°) = \frac{\sin(75°)}{10} \times 8 = 0.773$
Use the inverse sine function: $m = \sin^{-1}(0.773) = 50.6°$
Round to the nearest degree: 51°

🔍 Remember: When solving for angles, you'll need to use the inverse sine function $$\sin^{-1}$ or arcsin$ on your calculator to convert from the sine value back to degrees.

Practice Problems with the Sine Law

The best way to master the Sine Law is through practice! Let's look at some example problems that show different scenarios where the Sine Law comes in handy.

When finding an unknown angle like in problem 1, set up your proportion with the sine of your unknown angle on one side: $\frac{\sin(\theta)}{40} = \frac{\sin(55°)}{55}$ . Solving gives $\theta = 36.6°$ .

For finding an unknown side like in problem 2, use: $\frac{e}{\sin(49°)} = \frac{57}{\sin(59°)}$ . This gives e ≈ 50.2 m after calculation.

When solving a complete triangle (like problem 3), follow a strategic approach:

First find the missing angle using the fact that angles in a triangle sum to 180°
Then use the Sine Law to find remaining sides
Double-check your work by seeing if all parts fit together logically

⭐ Strategy Tip: When solving a triangle completely, first determine what you know (SSA, AAS, etc.), then use the most direct path to find missing values—sometimes this means finding an angle first, sometimes a side.

More Practice and Complex Applications

Ready to challenge yourself? Let's tackle some more complex applications of the Sine Law.

For problem 6a, we're looking for side a in a triangle where we know one side (30 ft) and two angles (39° and 65°). Using the Sine Law: $\frac{a}{\sin(39°)} = \frac{30}{\sin(65°)}$ , which gives us a ≈ 20.8 ft.

In problem 7, we're solving complete triangles. This requires a systematic approach:

First find any missing angles using the fact that angles in a triangle sum to 180°
Then use the Sine Law to find each unknown side
Organize your work step-by-step to avoid confusion

For triangle 7a with angles 57° and 38° and side 33 ft:

The third angle must be 85° $since 180° - 57° - 38° = 85°$
Side a ≈ 39.2 ft using $\frac{a}{\sin(85°)} = \frac{33}{\sin(57°)}$
Side c ≈ 24.2 ft using $\frac{c}{\sin(38°)} = \frac{33}{\sin(57°)}$

🧩 Problem-Solving Tip: When solving complete triangles, draw and label all the known information clearly first. This visual approach helps prevent errors and clarifies which formula to use.

Real-World Applications of the Sine Law

The Sine Law isn't just for math class—it helps solve real problems like finding distances and heights that can't be measured directly.

Problem 8 shows how the Sine Law connects to the basic sine ratio you already know. In a right triangle, finding x using $\sin(50°) = \frac{8}{x}$ gives x ≈ 10.4. The Sine Law approach $\frac{x}{\sin(90°)} = \frac{8}{\sin(50°)}$ yields the same result—proving these concepts are related!

Problem 9 tackles a practical scenario with a TV tower and guy wires. We can find both the tower height and distance between anchor points:

First, find the tower height using basic trigonometry: $h = 27\sin(32°) = 14.3$ m
Use the Sine Law to find the angle at point C: $\frac{\sin(\theta)}{27} = \frac{\sin(32°)}{15}$ , giving $\theta = 72.5°$
Find angle A using the fact that angles in a triangle sum to 180°: $75.5°$
Finally, find distance a using the Sine Law: $a = \frac{15\sin(75.5°)}{\sin(32°)} = 27.4$ m

🌟 Real-World Connection: Engineers and surveyors use the Sine Law regularly to calculate distances and angles that can't be measured directly—like determining the height of structures or distances across terrain that's difficult to traverse.

We thought you’d never ask...

What is the Knowunity AI companion?

Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.

Where can I download the Knowunity app?

You can download the app in the Google Play Store and in the 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.

4.9/5

App Store

4.8/5

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

I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️

Thomas R

iOS user

Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

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

I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. 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 app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!

Paul T

iOS user

Stefan S

iOS user

Samantha Klich

Android user

Anna

iOS user

Thomas R

iOS user

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

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

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

Elisha

iOS user

Paul T

iOS user

Trigonometry

•

Dec 8, 2025

•

5 pages

Mastering the Law of Sines: Explanation and Practice Problems

Chelsea

@zuncho

The Sine Law is a powerful tool for solving triangles that don't have right angles. Unlike the trig ratios you've used before, the Sine Law works on any triangle when you know a side and its opposite angle. This concept... Show more

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Understanding the Sine Law

Ever wondered how to solve triangles that aren't right-angled? The Sine Law is your answer! This formula creates a relationship between the sides of any triangle and their opposite angles.

The Sine Law can be written two ways depending on what you need to find:

For finding sides: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$
For finding angles: $\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}$

When using the Sine Law, remember that each side pairs with its opposite angle (side a faces angle A). You only need to work with two pairs at a time, not all three.

💡 Quick Tip: Think of the Sine Law as a proportion - if you know one side-angle pair, you can find another side or angle using that relationship.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Finding Missing Angles Using the Sine Law

Now let's flip things around and use the Sine Law to find missing angles. When you know two sides and an angle opposite one of them, you can discover the missing angle.

For finding angles, we rearrange the formula to: $\frac{\sin(A)}{a} = \frac{\sin(B)}{b}$ . This version puts the angles in the numerator position, making it easier to solve for an unknown angle.

Solving step by step:

Multiply both sides by 8: $\sin(m°) = \frac{\sin(75°)}{10} \times 8 = 0.773$
Use the inverse sine function: $m = \sin^{-1}(0.773) = 50.6°$
Round to the nearest degree: 51°

🔍 Remember: When solving for angles, you'll need to use the inverse sine function $$\sin^{-1}$ or arcsin$ on your calculator to convert from the sine value back to degrees.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Practice Problems with the Sine Law

The best way to master the Sine Law is through practice! Let's look at some example problems that show different scenarios where the Sine Law comes in handy.

For finding an unknown side like in problem 2, use: $\frac{e}{\sin(49°)} = \frac{57}{\sin(59°)}$ . This gives e ≈ 50.2 m after calculation.

When solving a complete triangle (like problem 3), follow a strategic approach:

First find the missing angle using the fact that angles in a triangle sum to 180°
Then use the Sine Law to find remaining sides
Double-check your work by seeing if all parts fit together logically

⭐ Strategy Tip: When solving a triangle completely, first determine what you know (SSA, AAS, etc.), then use the most direct path to find missing values—sometimes this means finding an angle first, sometimes a side.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

More Practice and Complex Applications

Ready to challenge yourself? Let's tackle some more complex applications of the Sine Law.

In problem 7, we're solving complete triangles. This requires a systematic approach:

First find any missing angles using the fact that angles in a triangle sum to 180°
Then use the Sine Law to find each unknown side
Organize your work step-by-step to avoid confusion

For triangle 7a with angles 57° and 38° and side 33 ft:

The third angle must be 85° $since 180° - 57° - 38° = 85°$
Side a ≈ 39.2 ft using $\frac{a}{\sin(85°)} = \frac{33}{\sin(57°)}$
Side c ≈ 24.2 ft using $\frac{c}{\sin(38°)} = \frac{33}{\sin(57°)}$

🧩 Problem-Solving Tip: When solving complete triangles, draw and label all the known information clearly first. This visual approach helps prevent errors and clarifies which formula to use.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Real-World Applications of the Sine Law

The Sine Law isn't just for math class—it helps solve real problems like finding distances and heights that can't be measured directly.

Problem 9 tackles a practical scenario with a TV tower and guy wires. We can find both the tower height and distance between anchor points:

First, find the tower height using basic trigonometry: $h = 27\sin(32°) = 14.3$ m
Use the Sine Law to find the angle at point C: $\frac{\sin(\theta)}{27} = \frac{\sin(32°)}{15}$ , giving $\theta = 72.5°$
Find angle A using the fact that angles in a triangle sum to 180°: $75.5°$
Finally, find distance a using the Sine Law: $a = \frac{15\sin(75.5°)}{\sin(32°)} = 27.4$ m

🌟 Real-World Connection: Engineers and surveyors use the Sine Law regularly to calculate distances and angles that can't be measured directly—like determining the height of structures or distances across terrain that's difficult to traverse.