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)

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