Learning Materials
Explanations
Features
Discover
Pythagorean identities are equations based on Pythagoras' theorem \( a^2 + b^2 = c^2\). You can use this theorem to find the sides of a right-angled triangle. There are three Pythagorean identities.
Millions of flashcards designed to help you ace your studies
Sign up for freeWhat is StudySmarter?
How does StudySmarter help me study more efficiently?
Where can I find more explanations like this?
What's smart about StudySmarter's flashcards?
Can I create my own content on StudySmarter?
How does spaced repetition work in StudySmarter flashcards?
What can you do with flashcards in StudySmarter?
Is StudySmarter a science-based learning platform?
How do StudySmarter's smart learning plans support your exam prep?
Can you create your own study sets in StudySmarter?
What is StudySmarter?
How does StudySmarter help me study more efficiently?
Where can I find more explanations like this?
What's smart about StudySmarter's flashcards?
Can I create my own content on StudySmarter?
How does spaced repetition work in StudySmarter flashcards?
What can you do with flashcards in StudySmarter?
Is StudySmarter a science-based learning platform?
How do StudySmarter's smart learning plans support your exam prep?
Can you create your own study sets in StudySmarter?
Achieve better grades quicker with Premium
Geld-zurück-Garantie, wenn du durch die Prüfung fällst
Did you know that StudySmarter supports you beyond learning?
Review generated flashcards
to start learning or create your own AI flashcards
Start learning or create your own AI flashcards
Team Pythagorean Identities Teachers
Jump to a key chapter
Right-angled triangle used for the base of Pythagoras theorem
The first Pythagorean identity is \( \sin^2 \theta + \cos^2 \theta = 1\). This can be derived using Pythagoras theorem and the unit circle.
Unit circle showing the derivation for the first Pythagorean identity
We know that \( a^2 + b^2 = c^2\) so \( \sin^2 \theta + \cos^2 \theta = 1\).
The second Pythagorean identity is \( \tan^2\theta + 1 = \sec^2\theta \). This is derived by taking the first Pythagorean identity and dividing it by \(\cos^2\theta\):
\[ \frac{\sin^2\theta}{\cos^2\theta} + \frac{\cos^2\theta} {\cos^2\theta} = \frac{1}{\cos^2\theta} .\]
Remember that
\[ \frac{\sin\theta}{\cos\theta} = \tan\theta \mbox{ and } \frac{1}{\cos\theta} = \sec\theta.\]
Simplifying this expression we get \( \tan^2\theta + 1 = \sec^2\theta \).
The third Pythagorean identity is \( 1 + \cot^2\theta = \csc^2\theta\). This is derived by taking the first Pythagorean identity and dividing it by \(\sin^2\theta\):
\[ \frac{\sin^2\theta}{\sin^2\theta} + \frac{\cos^2\theta} {\sin^2\theta} = \frac{1}{\sin^2\theta} .\]
Remember that
\[ \frac{\cos\theta}{\sin\theta} = \cot\theta \mbox{ and } \frac{1}{\sin\theta} = \csc\theta.\]
Now we can simplify this expression to \( 1 + \cot^2\theta = \csc^2\theta\).
We will now look at three examples of using each of the Pythagorean identities to answer questions.
Simplify \(\sin x \cos^2 x = \sin x -1\) and find the value of \(x\): \(0 ＜ x ＜ 2\pi\).
For this, we will need to use the first Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1\) and rearrange it:
\[ \cos^2 x = 1 - \sin^2 x .\]
We can now substitute \(1 - \sin^2 x \) into the expression:
\[ \sin x \cos^2 x = \sin x(1 - \sin^2 x ).\]
Simplifying this and setting it equal to the right hand side, we get
\[ \sin x - \sin^3 x = \sin x -1 \]
or
\[-\sin^3 x = -1. \]
So \( \sin x = 1 \) and \(x = \frac{\pi}{2}\).
If \(\cos x = 0.78\), what is the value of \(\tan x\)?
For this, we need to use the fact that \( \tan^2x + 1 = \sec^2x \). We also know that
\[ \sec x = \frac{1}{\cos x}\]
therefore
\[ \sec x = \frac{1}{0.78} = 1.282 .\]
We can now substitute this value into the equation and find \( \tan x\):
\[ \tan^2 x + 1 = (1.282)^2 \]
so
\[ \tan^2 x = (1.282)^2 -1 \]
and \( \tan x = 0.802\).
Solve for \(x\) between \(0^\circ\) and \(180^\circ\):
\[ \cot^2 (2x)+ \csc (2x) - 1 = 0.\]
In this case, we need to use the third Pythagorean identity, \( 1 + \cot^2\theta = \csc^2\theta\).
If we rearrange this identity, we get \( \cot^2\theta = \csc^2\theta - 1\). In this case \(\theta = 2x\) and we can plug in this rearranged identity into our equation:
\[ \left( \csc^2(2x) - 1 \right) + \csc 2x - 1 = 0 \]
so
\[ \csc^2 2x + \csc 2x - 2 = 0.\]
We can treat this as a quadratic that we can factorise into
\[(\csc 2x + 2)(\csc 2x - 1) = 0.\]
We can now solve this and get \( \csc 2x = -2\) or \( \csc 2x = 1\), so \( \sin 2x = -\frac{1}{2}\) or \(\sin x = 1\). Therefore \(2x = 210^\circ\), \(330^\circ\), \(90^\circ\). and \(x = 45^\circ\), \(105^\circ\), \(165^\circ\).
The first Pythagorean identity is \( \sin^2 \theta + \cos^2 \theta = 1\).
The second Pythagorean identity is \( \tan^2\theta + 1 = \sec^2\theta \).
The third Pythagorean identity is \( 1 + \cot^2\theta = \csc^2\theta\).
The first identity is derived from the Pythagorean theorem \( a^2 + b^2 = c^2\) and the unit circle.
The second and third identities are derived from the first identity.
Sign up for free to gain access to all our flashcards.
How do you derive Pythagorean identities?
The Pythagorean identities are derived from Pythagoras theorem and the unit circle.
What are Pythagorean identities?
They are expressions which are based on Pythagoras theorem and can be used to solve or simplify trigonometric equations.
What are the three Pythagorean identities?
sin^2(𝛉) +cos^2(𝛉) =1, tan^2(𝝷)+1=sec^2(𝝷) and 1+cot^2(𝝷)=csc^2(𝝷)
At StudySmarter, we have created a learning platform that serves millions of students. Meet the people who work hard to deliver fact based content as well as making sure it is verified.
Digital Content Specialist
Lily Hulatt is a Digital Content Specialist with over three years of experience in content strategy and curriculum design. She gained her PhD in English Literature from Durham University in 2022, taught in Durham University’s English Studies Department, and has contributed to a number of publications. Lily specialises in English Literature, English Language, History, and Philosophy.
Get to know Lily
AI Engineer
Gabriel Freitas is an AI Engineer with a solid experience in software development, machine learning algorithms, and generative AI, including large language models’ (LLMs) applications. Graduated in Electrical Engineering at the University of São Paulo, he is currently pursuing an MSc in Computer Engineering at the University of Campinas, specializing in machine learning topics. Gabriel has a strong background in software engineering and has worked on projects involving computer vision, embedded AI, and LLM applications.
Get to know Gabriel
StudySmarter is a globally recognized educational technology company, offering a holistic learning platform designed for students of all ages and educational levels. Our platform provides learning support for a wide range of subjects, including STEM, Social Sciences, and Languages and also helps students to successfully master various tests and exams worldwide, such as GCSE, A Level, SAT, ACT, Abitur, and more. We offer an extensive library of learning materials, including interactive flashcards, comprehensive textbook solutions, and detailed explanations. The cutting-edge technology and tools we provide help students create their own learning materials. StudySmarter’s content is not only expert-verified but also regularly updated to ensure accuracy and relevance.
Learn more
Team Math Teachers
Save explanations to your personalised space and access them anytime, anywhere!
Sign up with Email Sign up with AppleBy signing up, you agree to the Terms and Conditions and the Privacy Policy of StudySmarter.
Already have an account? Log in
Sign up to highlight and take notes. It’s 100% free.
Flashcards 
StudySmarter AI 
Notes 
Study Plans 
Study Sets 
Exams 
Find a job 
Student Deals 
Magazine 
Mobile App 
