Pythagorean Identities

Pythagorean Identities

The Pythagorean theorem can be applied to the trigonometric ratios that give rise to the Pythagorean identity. In this step-by-step guide, you will learn the concept of Pythagorean identity.

Pythagorean Identities

In mathematics, identity is an equation that holds for all possible values. An equation that contains trigonometric functions and is true for any value that replaces the variable is called trigonometric identity.

Related Topics

A step-by-step guide to Pythagorean identities

Pythagorean identities are important identities in trigonometry derived from the Pythagorean theorem. These identities are used to solve many trigonometric problems in which a trigonometric ratio is given and other ratios are found.

The fundamental Pythagorean identity shows the relationship between \(sin\) and \(cos\), and is the most common Pythagorean identity that says:

  • \(\color{blue}{sin^2\theta +cos^2\theta =1}\) (which gives the relation between \(sin\) and \(cos\))

There are two other Pythagorean identities as follows:

  • \(\color{blue}{sec^2\theta -tan^2\theta =1}\) (which gives the relation between \(sec\) and \(tan\))
  • \(\color{blue}{csc^2\theta -cot^2\theta =1}\) (which gives the relation between \(csc\) and \(cot\))

Pythagorean trig identities

All Pythagorean trig identities are listed below.

  • \(\color{blue}{sin^2\theta +cos^2\theta =1}\)
  • \(\color{blue}{1+tan^2\theta =sec^2\theta}\)
  • \(\color{blue}{1+cot^2\theta =cosec^2\theta}\)

Each of them can be written in different forms with algebraic operations. That is, any Pythagorean identity can be written in three ways as follows:

  • \(\color{blue}{sin^2θ + cos^2θ = 1 ⇒ 1 – sin^2θ = cos^2 θ ⇒ 1 – cos^2θ = sin^2θ}\)
  • \(\color{blue}{sec^2θ\ – tan^2θ = 1 ⇒ sec^2θ = 1 + tan^2θ ⇒ sec^2θ – 1 = tan^2θ}\)
  • \(\color{blue}{csc^2θ\ – cot^2θ = 1 ⇒ csc^2θ = 1 + cot^2θ ⇒ csc^2θ – 1 = cot^2θ}\)

Pythagorean Identities – Example 1:

In a right-angled triangle \(ABC\), angle \(C=90^{\circ }\), \(BAC = θ\), \(sin\:\theta = \frac{4}{5}\). Find the value of \(cos\:\theta\).

Solution:

Use the identity \(sin^2θ + cos^2θ =1\)

\((\frac{4}{5})^2+cos^2θ = 1\)

\(cos^2θ=1-(\frac{4}{5})^2\)

\(cos\:\theta ={\sqrt{1-\left(\frac{4}{5}\right)^2}}\)

\(=\sqrt{\frac{9}{25}}\)

\(=\frac{3}{5}\)

Exercises for Pythagorean Identities

  1. Suppose that \(sec\:\theta =\:-\frac{29}{20}\), what is the value of \(tan\:\theta\) if it is also negative?
  2. If \(sin\:\theta\) and \(cos\:\theta\) are the roots of the quadratic equation \(x^2+ px +1= 0\), find \(p\).
  3. If \(sin\:\theta \:cos\:\theta =\frac{1}{4}\), what is the value of \(sin\:\theta \:-\:cos\:\theta\)?

This image has an empty alt attribute; its file name is answers.png

  1. \(\color{blue}{-\frac{21}{20}}\)
  2. \(\color{blue}{\pm \sqrt{3}}\)
  3. \(\color{blue}{\frac{\sqrt{2}}{2}}\)

Related to This Article

More math articles

  • The Ultimate PSAT 8/9 Math Formula Cheat Sheet
  • Bеѕt Strategies to pass the Aссuрlасеr Test
  • 5 Best CHSPE Math Study Guides
  • How to Evaluate Recursive Formulas for Sequences
  • Algebra Puzzle – Critical Thinking 13
  • Top 10 3rd Grade NYSE Math Practice Questions
  • The Ultimate ACT Math Course (+FREE Worksheets & Tests)
  • FREE 5th Grade Common Core Math Practice Test
  • A Comprehensive Collection of FREE PSAT Math Practice Tests
  • 10 Most Common 5th Grade Georgia Milestones Assessment System Math Questions

What people say about “Pythagorean Identities – Effortless Math: We Help Students Learn to LOVE Mathematics”?

No one replied yet.

You are watching: Pythagorean Identities. Info created by THVinhTuy selection and synthesis along with other related topics.

Rate this post

Related Posts