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Layla Guthrie
12/3/2025
Pre-Calculus
Sketching sine and cosine graphs and inverse trig functions
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Dec 3, 2025
•
Understanding Sine and Cosine Graphs and Trigonometric Inverses
1 / 3
Sketching Graphs of Sine and Cosine with Period Changes
Ever wondered how changing the value inside sine or cosine affects its graph? When you see functions like y = sin(bx) or y = cos(bx), the value of b changes how quickly the function completes one full cycle.
The period of a trigonometric function is calculated using the formula 2π/b. For example, in y = sin(3x), the period is 2π/3, meaning the function completes a full cycle three times faster than the standard sine function. Similarly, for y = sin, the period becomes 4π/3.
To sketch these graphs, identify key points within one period. For cosine functions, these points typically include where the function equals 1, 0, -1, and back to 0. For sine functions, they're where the function equals 0, 1, 0, -1, and back to 0.
Pro Tip: When sketching trig graphs with modified periods, always calculate the new period first , then divide it into equal parts to find your key points. This makes the sketching process much more manageable!
Vertical Shifts and Amplitude Changes
Transforming trig graphs gets easier when you understand how different changes affect the output. When you add a constant to a function like y = cos(x) - 2, you're creating a vertical shift that moves the entire graph down by 2 units.
To sketch these transformed functions, create a table of values for the original function first, then apply the transformation. For vertical shifts, simply add or subtract the constant from each y-value. The shape remains the same, but the position changes.
When dealing with amplitude changes like y = -5/2 cos(2x), you multiply all y-values by the coefficient . This stretches or compresses the graph vertically, while the negative sign flips it upside down. Remember that the amplitude is the absolute value of this coefficient.
Don't forget to adjust the period too when dealing with functions like y = -5/2 cos(2x). The period changes to π , which means you'll need to recalculate your key x-values at 0, π/4, π/2, 3π/4, and π.
Remember: When sketching y = a·sin(bx) or y = a·cos(bx), the coefficient a affects the height (amplitude), while b affects the width (period) of the graph!
Phase Shifts and Inverse Trig Functions
Phase shifts occur when you have functions like y = 3cos, which shifts the cosine graph horizontally. To find the new x-coordinates, solve for when the inside expression equals your key values .
For example, to find where y = 3cos equals 3, solve x + 2π/3 = 0, giving x = -2π/3. Continue this process to find all key points, then sketch your graph using the amplitude of 3.
Inverse trigonometric functions find angles that produce specific trig values. The notation sin⁻¹(x) means "the angle whose sine equals x." These functions have restricted domains and ranges:
- sin⁻¹(x): Range is
- cos⁻¹(x): Range is
- tan⁻¹(x): Range is (-π/2, π/2)
The domains and ranges of the six trig functions are essential to understand. While sine and cosine are defined for all real numbers with ranges from -1 to 1, others like tangent and secant have restrictions. Tangent and cotangent have ranges of all real numbers, but their domains exclude specific values where they're undefined.
Quick Check: When evaluating sin⁻¹(-√3/2), remember you're finding the angle in whose sine equals -√3/2, which is -π/3.
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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
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
Understanding Sine and Cosine Graphs and Trigonometric Inverses
Dive into the world of trigonometric functions and their graphs! This guide will help you understand how to sketch various trig functions by manipulating amplitude, period, and phase shifts - essential skills for math success in algebra and precalculus.
Sketching Graphs of Sine and Cosine with Period Changes
Ever wondered how changing the value inside sine or cosine affects its graph? When you see functions like y = sin(bx) or y = cos(bx), the value of b changes how quickly the function completes one full cycle.
The period of a trigonometric function is calculated using the formula 2π/b. For example, in y = sin(3x), the period is 2π/3, meaning the function completes a full cycle three times faster than the standard sine function. Similarly, for y = sin, the period becomes 4π/3.
To sketch these graphs, identify key points within one period. For cosine functions, these points typically include where the function equals 1, 0, -1, and back to 0. For sine functions, they're where the function equals 0, 1, 0, -1, and back to 0.
Pro Tip: When sketching trig graphs with modified periods, always calculate the new period first , then divide it into equal parts to find your key points. This makes the sketching process much more manageable!
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Vertical Shifts and Amplitude Changes
Transforming trig graphs gets easier when you understand how different changes affect the output. When you add a constant to a function like y = cos(x) - 2, you're creating a vertical shift that moves the entire graph down by 2 units.
To sketch these transformed functions, create a table of values for the original function first, then apply the transformation. For vertical shifts, simply add or subtract the constant from each y-value. The shape remains the same, but the position changes.
When dealing with amplitude changes like y = -5/2 cos(2x), you multiply all y-values by the coefficient . This stretches or compresses the graph vertically, while the negative sign flips it upside down. Remember that the amplitude is the absolute value of this coefficient.
Don't forget to adjust the period too when dealing with functions like y = -5/2 cos(2x). The period changes to π , which means you'll need to recalculate your key x-values at 0, π/4, π/2, 3π/4, and π.
Remember: When sketching y = a·sin(bx) or y = a·cos(bx), the coefficient a affects the height (amplitude), while b affects the width (period) of the graph!
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Phase Shifts and Inverse Trig Functions
Phase shifts occur when you have functions like y = 3cos, which shifts the cosine graph horizontally. To find the new x-coordinates, solve for when the inside expression equals your key values .
For example, to find where y = 3cos equals 3, solve x + 2π/3 = 0, giving x = -2π/3. Continue this process to find all key points, then sketch your graph using the amplitude of 3.
Inverse trigonometric functions find angles that produce specific trig values. The notation sin⁻¹(x) means "the angle whose sine equals x." These functions have restricted domains and ranges:
- sin⁻¹(x): Range is
- cos⁻¹(x): Range is
- tan⁻¹(x): Range is (-π/2, π/2)
The domains and ranges of the six trig functions are essential to understand. While sine and cosine are defined for all real numbers with ranges from -1 to 1, others like tangent and secant have restrictions. Tangent and cotangent have ranges of all real numbers, but their domains exclude specific values where they're undefined.
Quick Check: When evaluating sin⁻¹(-√3/2), remember you're finding the angle in whose sine equals -√3/2, which is -π/3.
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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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
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