Trigonometric Functions and Identities
1.0 Trigonometric Ratios
1.0 Trigonometric Ratios
Relation between sides and angles of a right angled triangle are called trigonometric ratios or T-ratios. Trigonometric ratios can be represented in a right angled triangle $\Delta ABC$ as:
| $$\sin \theta = \frac{{Perpendicular}}{{Hypotenuese}} = \frac{{BC}}{{AC}}$$ | $${\text{cosec}}\theta = \frac{1}{{\sin \theta }} = \frac{{Hypotenuese}}{{Perpendicular}} = \frac{{AC}}{{BC}}$$ |
| $$cos\theta = \frac{{Base}}{{Hypotenuese}} = \frac{{AB}}{{AC}}$$ | $$sec\theta = \frac{1}{{\cos \theta }} = \frac{{Hypotenuese}}{{Base}} = \frac{{AC}}{{AB}}$$ |
| $$tan\theta = \frac{{Perpendicular}}{{Base}} = \frac{{BC}}{{AB}}$$ | $$cot\theta = \frac{1}{{\tan \theta }} = \frac{{Base}}{{Perpendicular}} = \frac{{AB}}{{BC}}$$ |
- To understand the angle $\theta$, let us consider a right angled triangle $\Delta ABC$. As we all know that a right angle triangle consists of three angles in which one angle is a right angle i.e., ${90^ \circ }$ and other two angles are always less than ${90^ \circ }$, call them as non-right angles. The side opposite to right angle is termed as "Hypotenuse" and we can say that the angle $\theta$ is always the angle between the hypotenuse and other lines. We termed the other line as "Base" and the third remaining line as "Perpendicular".
- The angle $\theta$ is measured in degree or radian.
- If a rotation from the initial side to terminal side is ${\left( {\frac{1}{{360}}} \right)^{th}}$ of a revolution, the angle is said to have a measure of one degree ${1^ \circ }$ as explained in figure.
- Angle subtended at the centre by an arc of length $1$ unit in a unit circle (circle of radius $1$ unit) is said to have a measure of $1$ radian.
- We know that the circumference of a circle of radius $1$ unit is $2\pi$. Thus, one complete revolution of the initial side subtends an angle of $2\pi$ radian and we can say that $$\begin{equation} \begin{aligned} 2\pi = {360^ \circ } \\ \Rightarrow \pi = {180^ \circ } \\ \therefore 1{\text{ radian}} = \frac{{{{180}^ \circ }}}{\pi } \\\end{aligned} \end{equation} $$
| Degree | $${30^ \circ }$$ | $${45^ \circ }$$ | $${60 ^ \circ }$$ | $${90^ \circ }$$ | $${180^ \circ }$$ | $${270^ \circ }$$ | $${360^ \circ }$$ |
| Radian | $$\frac{\pi }{6}$$ | $$\frac{\pi }{4}$$ | $$\frac{\pi }{3}$$ | $$\frac{\pi }{2}$$ | $$\pi $$ | $$\frac{3\pi }{2}$$ | $$2\pi $$ |
