Subjects
Biology
Chemistry
AP Chemistry
English
AP English Literature
Trigonometry
Dec 2, 2025
92
3 pages
Ready to solve real-world problems using trigonometry? These problems show how angles of elevation and depression help us... Show more
When you need to find the height of something tall without measuring it directly, trigonometry comes to the rescue! In problem 7, we used the tangent ratio to find a traffic light's height. When the angle of elevation is 30° from 30 meters away, we simply use tan 30° = height/30 to solve for the height.
For situations involving airplanes, like problem 8, we use the angle of depression (the downward angle from horizontal). When a plane flies at 12,000 meters and sees an airport tower at a 37° depression angle, we calculate the horizontal distance using tangent again.
Problem 9 shows a practical application with a ladder. When a 12-meter ladder leans against a wall at a 15° angle, we use sine to find the wall's height sin 15° = wall height/12. This gives us a wall height of 3 meters.
Quick Tip Remember that angles of elevation look up from horizontal, while angles of depression look down. Both types of problems use the same trigonometric ratios!
In problem 10, we find the minimum takeoff angle for an airplane to clear a building. By using the sine ratio with the building's height and the distance, we can determine the plane needs a 45° angle of elevation to safely clear the 63-foot building.
Parasailing provides a perfect example of trigonometry in action! In problem 11, we can find a parasailer's height when we know they're connected by a 200-foot line at a 53° angle using sine sin 53° = height/200. This gives us a height of 160 feet.
The sun's position creates shadows that change throughout the day. In problem 12, when a 10-meter tree casts a 17.3-meter shadow, we can determine the sun's angle of elevation using tangent tan x = 10/17.3. The calculation reveals the sun is 30° above the horizon.
Problem 13 presents a bird flying scenario where we need to find the horizontal distance. With the bird at 40 feet high, eyeing a perch on an 8-foot ledge at a 22° angle of depression, we calculate that the bird is 80 feet away horizontally.
Visualization Tip Draw these problems to scale on paper to help you see which ratio (sine, cosine, or tangent) makes the most sense for solving each problem.
In problem 14, we calculate a building's height using two angles. When Ms. Angel sees the building's top at a 14° angle of elevation and its base at a 67° angle of depression from 300 feet away, we find the total height by calculating both vertical distances separately. This gives us a surprisingly tall 795-foot building!
Rivers, platforms, and lighthouses offer interesting trigonometry challenges! In problem 15, Mr. Smith stands on a 36-foot cliff and sees two river banks at different angles of depression (53° and 37°). By calculating the distance to each bank and finding the difference, we determine the river is 21 feet wide.
Problem 16 involves two people at different positions. Jennie stands on a platform 15√3 meters high and sees the base of another platform at a 30° angle of depression. Meanwhile, Mark stands on the ground where Jennie sees him at a 60° angle of depression. By calculating both distances from the platform and finding their difference, we determine they're 30 meters apart.
Lighthouses were historically crucial for navigation, and problem 17 shows why! From a 75-meter lighthouse, an observer spots two ships at angles of depression of 30° and 45°. When both ships are in the same line from the lighthouse, we can calculate their horizontal distances and find the distance between them is 52.5 meters.
Real-world Connection These trigonometric calculations are essential in navigation, construction, astronomy, and engineering. Modern GPS systems use similar principles to determine locations!
Each of these problems follows a pattern identify what you know, determine the appropriate trigonometric ratio (usually sine, cosine, or tangent), set up an equation, and solve for the unknown value.
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.
You can download the app in the Google Play Store and in the Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
1
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
App Store
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
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
Ready to solve real-world problems using trigonometry? These problems show how angles of elevation and depression help us calculate distances and heights in everyday scenarios. With the right trigonometric ratio, you can find the height of buildings, distances between objects,... Show more
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Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
When you need to find the height of something tall without measuring it directly, trigonometry comes to the rescue! In problem 7, we used the tangent ratio to find a traffic light's height. When the angle of elevation is 30° from 30 meters away, we simply use tan 30° = height/30 to solve for the height.
For situations involving airplanes, like problem 8, we use the angle of depression (the downward angle from horizontal). When a plane flies at 12,000 meters and sees an airport tower at a 37° depression angle, we calculate the horizontal distance using tangent again.
Problem 9 shows a practical application with a ladder. When a 12-meter ladder leans against a wall at a 15° angle, we use sine to find the wall's height: sin 15° = wall height/12. This gives us a wall height of 3 meters.
Quick Tip: Remember that angles of elevation look up from horizontal, while angles of depression look down. Both types of problems use the same trigonometric ratios!
In problem 10, we find the minimum takeoff angle for an airplane to clear a building. By using the sine ratio with the building's height and the distance, we can determine the plane needs a 45° angle of elevation to safely clear the 63-foot building.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Parasailing provides a perfect example of trigonometry in action! In problem 11, we can find a parasailer's height when we know they're connected by a 200-foot line at a 53° angle using sine: sin 53° = height/200. This gives us a height of 160 feet.
The sun's position creates shadows that change throughout the day. In problem 12, when a 10-meter tree casts a 17.3-meter shadow, we can determine the sun's angle of elevation using tangent: tan x = 10/17.3. The calculation reveals the sun is 30° above the horizon.
Problem 13 presents a bird flying scenario where we need to find the horizontal distance. With the bird at 40 feet high, eyeing a perch on an 8-foot ledge at a 22° angle of depression, we calculate that the bird is 80 feet away horizontally.
Visualization Tip: Draw these problems to scale on paper to help you see which ratio (sine, cosine, or tangent) makes the most sense for solving each problem.
In problem 14, we calculate a building's height using two angles. When Ms. Angel sees the building's top at a 14° angle of elevation and its base at a 67° angle of depression from 300 feet away, we find the total height by calculating both vertical distances separately. This gives us a surprisingly tall 795-foot building!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Rivers, platforms, and lighthouses offer interesting trigonometry challenges! In problem 15, Mr. Smith stands on a 36-foot cliff and sees two river banks at different angles of depression (53° and 37°). By calculating the distance to each bank and finding the difference, we determine the river is 21 feet wide.
Problem 16 involves two people at different positions. Jennie stands on a platform 15√3 meters high and sees the base of another platform at a 30° angle of depression. Meanwhile, Mark stands on the ground where Jennie sees him at a 60° angle of depression. By calculating both distances from the platform and finding their difference, we determine they're 30 meters apart.
Lighthouses were historically crucial for navigation, and problem 17 shows why! From a 75-meter lighthouse, an observer spots two ships at angles of depression of 30° and 45°. When both ships are in the same line from the lighthouse, we can calculate their horizontal distances and find the distance between them is 52.5 meters.
Real-world Connection: These trigonometric calculations are essential in navigation, construction, astronomy, and engineering. Modern GPS systems use similar principles to determine locations!
Each of these problems follows a pattern: identify what you know, determine the appropriate trigonometric ratio (usually sine, cosine, or tangent), set up an equation, and solve for the unknown value.
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.
You can download the app in the Google Play Store and in the Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
1
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
App Store
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
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
