Introduction to trigonometry.

Trigonometric Ratios

Opposite & Adjacent Sides in a Right Angled Triangle

In the $\Delta ABC$ right-angled at B, BC is the side opposite to $\angle A,$ AC is the hypotenuse and AB is the side adjacent to $\angle A.$

Trigonometric Ratios

For the right $\Delta ABC$, right angled at $\angle B$, the trigonometric ratios of the $\angle A$ are as follows:

• $sinA=\frac{oppositeside}{hypotenuse}=\frac{BC}{AC}$
• $cosA=\frac{adjacentside}{hypotenuse}=\frac{AB}{AC}$
• $tanA=\frac{oppositeside}{adjacentside}=\frac{BC}{AB}$
• $cosecA=\frac{hypotenuse}{oppositeside}=\frac{AC}{BC}$
• $secA=\frac{hypotenuse}{adjacentside}=\frac{AC}{AB}$
• $cotA=\frac{adjacentside}{oppositeside}=\frac{AB}{BC}$

Visualisation of Trigonometric Ratios Using a Unit Circle

Draw a circle of unit radius with the origin as the centre. Consider a line segment OP joining a point P on the circle to the centre which makes an angle $\theta$ with the x-axis. Draw a perpendicular from P to the x-axis to cut it at Q.

• $sin\theta =\frac{PQ}{OP}=\frac{PQ}{1}=PQ$
• $cos\theta =\frac{OQ}{OP}=\frac{OQ}{1}=OQ$
• $tan\theta =\frac{PQ}{OQ}=\frac{sin\theta }{cos\theta }$
• $cosec\theta =\frac{OP}{PQ}=\frac{1}{PQ}$
• $sec\theta =\frac{OP}{OQ}=\frac{1}{OQ}$
• $cot\theta =\frac{OQ}{PQ}=\frac{cos\theta }{sin\theta }$

Visualisation of Trigonometric Ratios Using a Unit Circle

Relation between Trigonometric Ratios

• $cosec\theta =\frac{1}{sin\theta }$
• $sec\theta =\frac{1}{cos\theta }$
• $tan\theta =\frac{sin\theta }{cos\theta }$
• $cot\theta =\frac{cos\theta }{sin\theta }=\frac{1}{tan\theta }$

Trigonometric Ratios of Specific Angles

Range of Trigonometric Ratios from 0 to 90 degrees

For $0^{\circ }\le \theta \le {90}^{\circ },$

• $0\le sin\theta \le 1$
• $0\le cos\theta \le 1$
• $0\le tan\theta <\infty$
• $1\le sec\theta <\infty$
• $0\le cot\theta <\infty$
• $1\le cosec\theta <\infty$

$tan\theta$ and $sec\theta$ are not defined at  ${90}^{\circ }.$
$cot\theta$ and $cosec\theta$ are not defined at $0^{\circ }.$

Variation of trigonometric ratios from 0 to 90 degrees

As $\theta$ increases from $0^{\circ }$ to ${90}^{\circ }$

• $sin\theta$ increases from $0$ to $1$.
• $cos\theta$ decreases from $1$ to $0$.
• $tan\theta$ increases from $0$ to $\infty$.
• $cosec\theta$ decreases from $\infty$ to $1$.
• $sec\theta$ increases from $1$ to $\infty$.
• $cot\theta$ decreases from $\infty$ to $0$.

Standard values of Trigonometric ratios

$\begin{array}{cccccc}\angle A& 0^{\circ }& {30}^{\circ }& {45}^{\circ }& {60}^{\circ }& {90}^{\circ }\\ sinA& 0& \frac{1}{2}& \frac{1}{\sqrt{2}}& \frac{\sqrt{3}}{2}& 1\\ cosA& 1& \frac{\sqrt{3}}{2}& \frac{1}{\sqrt{2}}& \frac{1}{2}& 0\\ tanA& 0& \frac{1}{\sqrt{3}}& 1& \sqrt{3}& Notdefined\\ cosecA& Notdefined& 2& \sqrt{2}& \frac{2}{\sqrt{3}}& 1\\ secA& 1& \frac{2}{\sqrt{3}}& \sqrt{2}& 2& Notdefined\\ cotA& Notdefined& \sqrt{3}& 1& \frac{1}{\sqrt{3}}& 0\end{array}$

Trigonometric Ratios of Complementary Angles

Complementary Trigonometric ratios

If $\theta$ is an acute angle, its complementary angle is ${90}^{\circ }-\theta .$ The following relations hold true for trigonometric ratios of complementary angles.

• $sin({90}^{\circ }-\theta )=cos\theta$
• $cos({90}^{\circ }-\theta )=sin\theta$
• $tan({90}^{\circ }-\theta )=cot\theta$
• $cot({90}^{\circ }-\theta )=tan\theta$
• $cosec({90}^{\circ }-\theta )=sec\theta$
• $sec({90}^{\circ }-\theta )=cosec\theta$

Trigonometric Identities

Trigonometric Identities

• $si{n}^2\theta +co{s}^2\theta =1$
• $1+co{t}^2\theta =coes{c}^2\theta$
• $1+ta{n}^2\theta =se{c}^2\theta$