INTRODUCTION TO TRIGONOMETRY
IMPORTANT FORMULAS \& CONCEPTS
The word ‘trigonometry’ is derived from the Greek words ‘tri’ (meaning three), ‘gon’ (meaning sides) and ‘metron’ (meaning measure). In fact, trigonometry is the study of relationships between the sides and angles of a triangle.
\begin{aligned}
& \mathrm{P}(x, y) \\
& \sin \theta=\frac{\text { Opposite side }}{\text { Hypotenuse }}=\frac{y}{r} \quad \operatorname{cosec} \theta=\frac{\text { Hypotenuse }}{\text { Opposite side }}=\frac{r}{y} \\
& \cos \theta=\frac{\text { Adjacent Side }}{\text { Hypotenuse }}=\frac{x}{r} \quad \sec \theta=\frac{\text { Hypotenuse }}{\text { Adjacent Side }}=\frac{r}{x} \\
& \tan \theta=\frac{\text { Opposite side }}{\text { Adjacent Side }}=\frac{y}{x} \quad \cot \theta=\frac{\text { Adjacent Side }}{\text { Opposite side }}=\frac{x}{y} \\
& \text { Reciprocal Relations } \\
& \sin \theta=\frac{1}{\operatorname{cosec} \theta} \quad \operatorname{cosec} \theta=\frac{1}{\sin \theta} \\
& \cos \theta=\frac{1}{\sec \theta} \quad \sec \theta=\frac{1}{\cos \theta} \\
& \tan \theta=\frac{1}{\cot \theta} \quad \cot \theta=\frac{1}{\tan \theta} \\
& \text { Quotient Relations } \\
& \tan \theta=\frac{\sin \theta}{\cos \theta} \text { and } \cot \theta=\frac{\cos \theta}{\sin \theta}
\end{aligned}
TRIGONOMETRIC IDENTITIES
An equation involving trigonometric ratios of an angle is said to be a trigonometric identity if it is satisfied for all values of $$\theta$$ for which the given trigonometric ratios are defined.
$$
\begin{aligned}
\text { Identity (1) : } & \sin ^2 \theta+\cos ^2 \theta=1 \\
& \Rightarrow \sin ^2 \theta=1-\cos ^2 \theta \text { and } \cos ^2 \theta=1-\sin ^2 \theta \\
\text { Identity (2): } & \sec ^2 \theta=1+\tan ^2 \theta \\
& \Rightarrow \sec ^2 \theta-\tan ^2 \theta=1 \text { and } \tan ^2 \theta=\sec ^2 \theta-1 . \\
\text { Identity (3): } & \operatorname{cosec}^2 \theta=1+\cot ^2 \theta \\
& \Rightarrow \operatorname{cosec}^2 \theta-\cot ^2 \theta=1 \text { and } \cot ^2 \theta=\operatorname{cosec}^2 \theta-1 .
\end{aligned}
$$
Trigonometric ratios of Complementary angles.
$$\sin (90-\theta)=\cos \theta$$
$$\cos (90-\theta)=\sin \theta$$
$$\tan (90-\theta)=\cot \theta$$
$$\cot (90-\theta)=\tan \theta$$
$$\sec (90-\theta)=\operatorname{cosec} \theta$$
$$\operatorname{cosec}(90-\theta)=\sec \theta$$.
