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.
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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.
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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
- Suppose that \(sec\:\theta =\:-\frac{29}{20}\), what is the value of \(tan\:\theta\) if it is also negative?
- If \(sin\:\theta\) and \(cos\:\theta\) are the roots of the quadratic equation \(x^2+ px +1= 0\), find \(p\).
- 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](https://www.effortlessmath.com/wp-content/uploads/2019/12/answers.png)
- \(\color{blue}{-\frac{21}{20}}\)
- \(\color{blue}{\pm \sqrt{3}}\)
- \(\color{blue}{\frac{\sqrt{2}}{2}}\)
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