Basic Identities

6.4 Basic Identities
 
This image features three mathematical trigonometric identities presented in a stylized format. The identities are: 1. \( \sin^2 \theta + \cos^2 \theta = 1 \) 2. \( 1 + \tan^2 \theta = \sec^2 \theta \) 3. \( 1 + \cot^2 \theta = \csc^2 \theta \) These identities are displayed in speech bubble-like shapes, with arrows pointing towards the text 'BASIC IDENTITIES' at the bottom center. The 'Pandai' logo is also present below the text.
 
Derivation of Basic Identities Using Right-angled Triangles
Figure

A right-angled triangle labeled ABC, highlighting the right side of the triangle for clarity and emphasis.

Basic Identities
  • \(\sin{A}=\dfrac{a}{c}\)\(\cosec{A}=\dfrac{c}{a}\)
     
  • \(\cos{A}=\dfrac{b}{c}\)\(\sec{A}=\dfrac{c}{b}\)
     
  • \(\tan{A}=\dfrac{a}{b}\)\(\cot{A}=\dfrac{b}{a}\)
 
Derivation of Basic Identities Using the Pythagoras Theorem

By using Pythagoras theorem, it is known that \(a^2+b^2=c^2\). Divide the two sides of the equation by \(a^2\)\(b^2\) and \(c^2\), we get:

\(\div a^2\) \(\div b^2\) \(\div c^2\)

\(\dfrac{a^2}{a^2}+\dfrac{b^2}{a^2}=\dfrac{c^2}{a^2}\)

\(1+\left( \dfrac{b}{a} \right)^2=\left( \dfrac{c}{a} \right)^2\)

\(1+\cot^2{A}=\cosec^2{A}\)

\(\dfrac{a^2}{b^2}+\dfrac{b^2}{b^2}=\dfrac{c^2}{b^2}\)

\(\left( \dfrac{a}{b} \right)^2+1=\left( \dfrac{c}{b} \right)^2\)

\(1+\tan^2{A}=\sec^2{A}\)

\(\dfrac{a^2}{c^2}+\dfrac{b^2}{c^2}=\dfrac{c^2}{c^2}\)

\(\left( \dfrac{a}{c} \right)^2+\left( \dfrac{b}{c} \right)^2=1\)

\(\sin^2{A}+\cos^2{A}=1\)

 
Example
Question

Without using a calculator, find the value of each of the following:

(a) \(\text{sin}^2 (-430^{\circ}) + \text{cos}^2(-430^{\circ}) \)
(b) \(\text{tan}^2 \begin{pmatrix} \dfrac{\pi}{3} \end{pmatrix} - \text{sec}^2\begin{pmatrix} \dfrac{\pi}{3} \end{pmatrix} \)
Solution

(a)

\(\text{sin}^2 \ A + \text{cos}^2 \ A=1\\ \text{sin}^2 (-430^{\circ}) + \text{cos}^2(-430^{\circ}) =1.\)


(b)

\(\begin{aligned} &1+ \text{tan}^2 \ A = \text{sec}^2 \ A\\\\ &\text{tan}^2 \begin{pmatrix} \dfrac{\pi}{3} \end{pmatrix} - \text{sec}^2\begin{pmatrix} \dfrac{\pi}{3} \end{pmatrix} =-1 .\end{aligned}\)

 

Basic Identities

6.4 Basic Identities
 
This image features three mathematical trigonometric identities presented in a stylized format. The identities are: 1. \( \sin^2 \theta + \cos^2 \theta = 1 \) 2. \( 1 + \tan^2 \theta = \sec^2 \theta \) 3. \( 1 + \cot^2 \theta = \csc^2 \theta \) These identities are displayed in speech bubble-like shapes, with arrows pointing towards the text 'BASIC IDENTITIES' at the bottom center. The 'Pandai' logo is also present below the text.
 
Derivation of Basic Identities Using Right-angled Triangles
Figure

A right-angled triangle labeled ABC, highlighting the right side of the triangle for clarity and emphasis.

Basic Identities
  • \(\sin{A}=\dfrac{a}{c}\)\(\cosec{A}=\dfrac{c}{a}\)
     
  • \(\cos{A}=\dfrac{b}{c}\)\(\sec{A}=\dfrac{c}{b}\)
     
  • \(\tan{A}=\dfrac{a}{b}\)\(\cot{A}=\dfrac{b}{a}\)
 
Derivation of Basic Identities Using the Pythagoras Theorem

By using Pythagoras theorem, it is known that \(a^2+b^2=c^2\). Divide the two sides of the equation by \(a^2\)\(b^2\) and \(c^2\), we get:

\(\div a^2\) \(\div b^2\) \(\div c^2\)

\(\dfrac{a^2}{a^2}+\dfrac{b^2}{a^2}=\dfrac{c^2}{a^2}\)

\(1+\left( \dfrac{b}{a} \right)^2=\left( \dfrac{c}{a} \right)^2\)

\(1+\cot^2{A}=\cosec^2{A}\)

\(\dfrac{a^2}{b^2}+\dfrac{b^2}{b^2}=\dfrac{c^2}{b^2}\)

\(\left( \dfrac{a}{b} \right)^2+1=\left( \dfrac{c}{b} \right)^2\)

\(1+\tan^2{A}=\sec^2{A}\)

\(\dfrac{a^2}{c^2}+\dfrac{b^2}{c^2}=\dfrac{c^2}{c^2}\)

\(\left( \dfrac{a}{c} \right)^2+\left( \dfrac{b}{c} \right)^2=1\)

\(\sin^2{A}+\cos^2{A}=1\)

 
Example
Question

Without using a calculator, find the value of each of the following:

(a) \(\text{sin}^2 (-430^{\circ}) + \text{cos}^2(-430^{\circ}) \)
(b) \(\text{tan}^2 \begin{pmatrix} \dfrac{\pi}{3} \end{pmatrix} - \text{sec}^2\begin{pmatrix} \dfrac{\pi}{3} \end{pmatrix} \)
Solution

(a)

\(\text{sin}^2 \ A + \text{cos}^2 \ A=1\\ \text{sin}^2 (-430^{\circ}) + \text{cos}^2(-430^{\circ}) =1.\)


(b)

\(\begin{aligned} &1+ \text{tan}^2 \ A = \text{sec}^2 \ A\\\\ &\text{tan}^2 \begin{pmatrix} \dfrac{\pi}{3} \end{pmatrix} - \text{sec}^2\begin{pmatrix} \dfrac{\pi}{3} \end{pmatrix} =-1 .\end{aligned}\)