Trigonometry isn't just about angles and triangles—it's a powerful tool... Show more
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Solving Problems Using Trigonometry
Ever wondered how to approach those tricky trigonometry word problems? It's easier than you think with the right approach! Start by following these key steps:
- Read carefully and underline important information
- Identify what's given and what you need to find
- Use the given information to find the unknown values
- Sometimes you'll need an intermediate step before finding your final answer
When solving problems, ask yourself these questions to determine which method to use: Can I find angles using the triangle sum $180^\circ$? Do I need to make conversions? Can I use the Pythagorean Theorem? Should I use trigonometric ratios , Sine Law, or Cosine Law?
Pro Tip: Use this simple decision flowchart: If it's a right triangle, use trig ratios or Pythagorean theorem. If it's not a right triangle but you know a side and its opposite angle, use Sine Rule. Otherwise, use Cosine Rule.
Trigonometry Application Problems
Let's put trigonometry to work! In problem-solving, you'll need to round angles to the nearest degree and lengths to the nearest tenth (unless told otherwise).
When a flagpole casts a shadow 17.7 m long with the sun at an elevation of 66.4°, you can find the flagpole's height using the tangent ratio. Since this creates a right triangle, you'd use TOA : tan 66.4° = h/17.7, giving a height of 40.5 m.
For more complex shapes like the Bermuda Triangle , you'll need the cosine law to find the angles. This gives angles of 64.5° at Miami, 55.6° at Bermuda, and 60.4° at Puerto Rico.
Remember: Always draw a diagram for each problem, even if one isn't provided. This visual representation makes it much easier to identify which trigonometric method to use!
More Trigonometry Applications
Stage lighting, ferry routes, and hot air balloons all benefit from trigonometry! For example, when calculating the distance between two stage lights with known angles (66° and 89°) and a third light 28 m away, you'd use the sine law to find the missing distance of 60.5 m.
In the ferry problem, we have two hotels 550 m apart with angles of 49° and 56° from the dock. Using the sine law and knowing the total angle is 180°, we can determine that the dock is 472.1 m from Hotel A and 429.7 m from Hotel B.
For the hot air balloon problem, two observers 325 m apart measure elevation angles of 54° and 38°. The sine law helps us calculate that Seema is 263.1 m from the balloon and Hassim is 200.2 m away. The balloon's height can then be calculated using the sine ratio: h = 200.2 × sin 54° = 162 m.
Got stuck? If you're unsure which law to use, remember that the sine law works best when you have angles and their opposite sides, while the cosine law is perfect when you have a triangle with three sides or two sides and the included angle.
Advanced Trigonometry Problems
Ready to tackle more challenging scenarios? These real-world applications show trigonometry's incredible versatility!
For a bridge problem, when one end is 112 m from railroad tracks with a 37° angle of depression and the bridge is 122 m long, you can use the cosine law to determine that the other end is 75 m from the tracks.
Ancient structures like the Great Pyramid of Giza can be analyzed using trigonometry. With a base length of 230 m and an elevation angle of 52°, you can calculate the slant side length by applying the sine law.
Sports scenarios also rely on trigonometry. A golfer hitting a ball at a 64° angle needs to determine if it will clear a 30 ft tree 7 yards away. By using the tangent ratio, you can verify if the ball's height at that distance will exceed the tree's height.
Challenge yourself: When working with multiple angles and distances, try setting up a coordinate system to help visualize how the different measurements relate to each other. This approach can make complex problems much more manageable!
Solving Miscellaneous Trigonometry Problems
Trigonometry helps solve everyday problems across many fields! For a cone-shaped funnel with 15 cm sides and a 17.9° angle between sides, you can find the diameter using the cosine law: d² = 15² + 15² - 2(15)(15)cos17.9°, giving you approximately 4.7 cm.
Building height measurements become simple with trigonometry. If you stand 10 m from a building base and measure a 65° elevation angle from 6 ft (1.8 m) above ground, you can calculate the building's height using tangent: 10 × tan65° = 21 m, plus your height of 1.8 m, giving a total of approximately 23 m.
For the Great Pyramid problem, using the sine law with a base length of 230 m and elevation angle of 52° gives a slant side length of 186.8 m. Similarly, you can determine if a golf ball hit at 64° will clear a 30 ft tree 7 yards away by calculating that the ball reaches a height of 14.4 yards at that distance, which exceeds 10 yards (30 ft).
Real-world application: Engineers, architects, astronomers, and navigators all rely on these exact trigonometric methods in their daily work. The skills you're learning now are the same ones professionals use to design buildings, navigate ships, and explore space!
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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
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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
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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
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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❗️❗️⚠️
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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 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 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
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 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 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
Mastering Trigonometry: Key Concepts and Practice Problems
Trigonometry isn't just about angles and triangles—it's a powerful tool for solving real-world problems. This guide breaks down how to tackle trigonometry word problems step-by-step, showing you exactly when to use sine law, cosine law, or the Pythagorean theorem.
Solving Problems Using Trigonometry
Ever wondered how to approach those tricky trigonometry word problems? It's easier than you think with the right approach! Start by following these key steps:
- Read carefully and underline important information
- Identify what's given and what you need to find
- Use the given information to find the unknown values
- Sometimes you'll need an intermediate step before finding your final answer
When solving problems, ask yourself these questions to determine which method to use: Can I find angles using the triangle sum $180^\circ$? Do I need to make conversions? Can I use the Pythagorean Theorem? Should I use trigonometric ratios , Sine Law, or Cosine Law?
Pro Tip: Use this simple decision flowchart: If it's a right triangle, use trig ratios or Pythagorean theorem. If it's not a right triangle but you know a side and its opposite angle, use Sine Rule. Otherwise, use Cosine Rule.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
Trigonometry Application Problems
Let's put trigonometry to work! In problem-solving, you'll need to round angles to the nearest degree and lengths to the nearest tenth (unless told otherwise).
When a flagpole casts a shadow 17.7 m long with the sun at an elevation of 66.4°, you can find the flagpole's height using the tangent ratio. Since this creates a right triangle, you'd use TOA : tan 66.4° = h/17.7, giving a height of 40.5 m.
For more complex shapes like the Bermuda Triangle , you'll need the cosine law to find the angles. This gives angles of 64.5° at Miami, 55.6° at Bermuda, and 60.4° at Puerto Rico.
Remember: Always draw a diagram for each problem, even if one isn't provided. This visual representation makes it much easier to identify which trigonometric method to use!
Sign up to see the contentIt's free!
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More Trigonometry Applications
Stage lighting, ferry routes, and hot air balloons all benefit from trigonometry! For example, when calculating the distance between two stage lights with known angles (66° and 89°) and a third light 28 m away, you'd use the sine law to find the missing distance of 60.5 m.
In the ferry problem, we have two hotels 550 m apart with angles of 49° and 56° from the dock. Using the sine law and knowing the total angle is 180°, we can determine that the dock is 472.1 m from Hotel A and 429.7 m from Hotel B.
For the hot air balloon problem, two observers 325 m apart measure elevation angles of 54° and 38°. The sine law helps us calculate that Seema is 263.1 m from the balloon and Hassim is 200.2 m away. The balloon's height can then be calculated using the sine ratio: h = 200.2 × sin 54° = 162 m.
Got stuck? If you're unsure which law to use, remember that the sine law works best when you have angles and their opposite sides, while the cosine law is perfect when you have a triangle with three sides or two sides and the included angle.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
Advanced Trigonometry Problems
Ready to tackle more challenging scenarios? These real-world applications show trigonometry's incredible versatility!
For a bridge problem, when one end is 112 m from railroad tracks with a 37° angle of depression and the bridge is 122 m long, you can use the cosine law to determine that the other end is 75 m from the tracks.
Ancient structures like the Great Pyramid of Giza can be analyzed using trigonometry. With a base length of 230 m and an elevation angle of 52°, you can calculate the slant side length by applying the sine law.
Sports scenarios also rely on trigonometry. A golfer hitting a ball at a 64° angle needs to determine if it will clear a 30 ft tree 7 yards away. By using the tangent ratio, you can verify if the ball's height at that distance will exceed the tree's height.
Challenge yourself: When working with multiple angles and distances, try setting up a coordinate system to help visualize how the different measurements relate to each other. This approach can make complex problems much more manageable!
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
Solving Miscellaneous Trigonometry Problems
Trigonometry helps solve everyday problems across many fields! For a cone-shaped funnel with 15 cm sides and a 17.9° angle between sides, you can find the diameter using the cosine law: d² = 15² + 15² - 2(15)(15)cos17.9°, giving you approximately 4.7 cm.
Building height measurements become simple with trigonometry. If you stand 10 m from a building base and measure a 65° elevation angle from 6 ft (1.8 m) above ground, you can calculate the building's height using tangent: 10 × tan65° = 21 m, plus your height of 1.8 m, giving a total of approximately 23 m.
For the Great Pyramid problem, using the sine law with a base length of 230 m and elevation angle of 52° gives a slant side length of 186.8 m. Similarly, you can determine if a golf ball hit at 64° will clear a 30 ft tree 7 yards away by calculating that the ball reaches a height of 14.4 yards at that distance, which exceeds 10 yards (30 ft).
Real-world application: Engineers, architects, astronomers, and navigators all rely on these exact trigonometric methods in their daily work. The skills you're learning now are the same ones professionals use to design buildings, navigate ships, and explore space!
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.
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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 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 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
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 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 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