Trigonometry means the science of measurement of triangles.

Trigonometrical Ratios: This is the first thing that we need to ensure that we understand. The basic trigonometrical ratios are the following (refer to the diagram as shown):

$\displaystyle \text{(i) } \sin \theta = \frac{BC}{AC} \hspace{1.0cm} \text{(ii) } \cos \theta = \frac{AB}{AC} \hspace{1.0cm} \text{(iii) } \tan \theta = \frac{BC}{AB}$

$\displaystyle \text{(iv) } \mathrm{cosec} \theta = \frac{AC}{BC} \hspace{1.0cm} \text{(v) } \sec \theta = \frac{AC}{AB} \hspace{1.0cm} \text{(vi) } \cot \theta = \frac{AB}{BC}$

Notes:

• The trigonometrical ratio is a real number and has no unit.
• The values of the trigonometrical ratios will always remain the same. It depends on the angle and not on the size of the triangle.

Relation between trigonometrical ratios:

Reciprocal Relations:

$\displaystyle \text{(i) } \sin \theta = \frac{1}{\mathrm{cosec} \theta} \hspace{1.0cm} \text{(ii) } \cos \theta = \frac{1}{\sec \theta} \hspace{1.0cm} \text{(iii) } \tan \theta = \frac{1}{\cot \theta}$

Quotient Relations:

$\displaystyle \text{(i) } \tan \theta = \frac{\sin \theta}{\cos \theta} \hspace{1.0cm} \text{(ii) } \cot \theta = \frac{\cos \theta}{\sin \theta}$

Square Relations:

$\displaystyle \text{(i) } \sin^2 \theta + \cos^2 \theta = 1$

Proof:

$\displaystyle \sin^2 \theta + \cos^2 \theta = \Bigg(\frac{BC}{AC} \Bigg)^2 + \Bigg(\frac{AB}{AC} \Bigg)^2 = \frac{BC^2+AB^2}{AC^2} = \frac{AC^2}{AC^2} = 1$

$\displaystyle \text{(ii) } 1+ \tan^2 \theta = \sec^2 \theta$

Proof:

$\displaystyle 1+ \tan^2 \theta = 1 + \Bigg(\frac{BC}{AB} \Bigg)^2 = \frac{BC^2+AB^2}{AB^2} = \frac{AC^2}{AB^2} = \sec^2 \theta$

$\displaystyle \text{(iii) } 1+ \cot^2 \theta = \mathrm{cosec}^2 \theta$

Proof:

$\displaystyle 1+ \cot^2 \theta = 1 + \Bigg(\frac{AB}{BC} \Bigg)^2 = \frac{BC^2+AB^2}{BC^2} = \frac{AC^2}{BC^2} = \mathrm{cosec}^2 \theta$

Trigonometrical Ratios of complementary angles:

For $\displaystyle \theta < 90^{\circ}$

$\displaystyle \text{(i) } \sin (90^{\circ} - \theta) = \cos \theta \hspace{1.0cm} \text{(ii) } \cos (90^{\circ} - \theta) = \sin \theta$

$\displaystyle \text{(iii) } \tan (90^{\circ} - \theta) = \cot \theta \hspace{1.0cm} \text{(iv) } \cot (90^{\circ} - \theta) = \tan \theta$

$\displaystyle \text{(v) } \sec (90^{\circ} - \theta) = \mathrm{cosec} \theta \hspace{1.0cm} \text{(vi) } \mathrm{cosec} (90^{\circ} - \theta) = \sec \theta$

Using the Trigonometrical Tables:

Trigonometric Table in Sexagesimal System:

 Angles in Degrees $(\theta)$$(\theta)$ $0^o$$0^o$ $30^o$$30^o$ $45^o$$45^o$ $60^o$$60^o$ $90^o$$90^o$ $\sin$$\sin$ $0$$0$ $\frac{1}{2}$$\frac{1}{2}$ $\frac{1}{\sqrt{2}}$$\frac{1}{\sqrt{2}}$ $\frac{\sqrt{3}}{2}$$\frac{\sqrt{3}}{2}$ $1$$1$ $\cos$$\cos$ $1$$1$ $\frac{\sqrt{3}}{2}$$\frac{\sqrt{3}}{2}$ $\frac{1}{\sqrt{2}}$$\frac{1}{\sqrt{2}}$ $\frac{1}{2}$$\frac{1}{2}$ $0$$0$ $\tan$$\tan$ $0$$0$ $\frac{1}{\sqrt{3}}$$\frac{1}{\sqrt{3}}$ $1$$1$ $\sqrt{3}$$\sqrt{3}$ Not Defined $\mathrm{cosec}$$\mathrm{cosec}$ Not Defined $2$$2$ $\sqrt{2}$$\sqrt{2}$ $\frac{2}{\sqrt{3}}$$\frac{2}{\sqrt{3}}$ $1$$1$ $\sec$$\sec$ $1$$1$ $\frac{2}{\sqrt{3}}$$\frac{2}{\sqrt{3}}$ $\sqrt{2}$$\sqrt{2}$ $2$$2$ Not Defined $\cot$$\cot$ Not Defined $\sqrt{3}$$\sqrt{3}$ $1$$1$ $\frac{1}{\sqrt{3}}$$\frac{1}{\sqrt{3}}$ $0$$0$

Trigonometric Table in Circular System:

 Angles in Degrees $(\theta)$$(\theta)$ $0$$0$ $\frac{\pi}{6}$$\frac{\pi}{6}$ $\frac{\pi}{4}$$\frac{\pi}{4}$ $\frac{\pi}{3}$$\frac{\pi}{3}$ $\frac{\pi}{2}$$\frac{\pi}{2}$ $\sin$$\sin$ $0$$0$ $\frac{1}{2}$$\frac{1}{2}$ $\frac{1}{\sqrt{2}}$$\frac{1}{\sqrt{2}}$ $\frac{\sqrt{3}}{2}$$\frac{\sqrt{3}}{2}$ $1$$1$ $\cos$$\cos$ $1$$1$ $\frac{\sqrt{3}}{2}$$\frac{\sqrt{3}}{2}$ $\frac{1}{\sqrt{2}}$$\frac{1}{\sqrt{2}}$ $\frac{1}{2}$$\frac{1}{2}$ $0$$0$ $\tan$$\tan$ $0$$0$ $\frac{1}{\sqrt{3}}$$\frac{1}{\sqrt{3}}$ $1$$1$ $\sqrt{3}$$\sqrt{3}$ Not Defined $\mathrm{cosec}$$\mathrm{cosec}$ Not Defined $2$$2$ $\sqrt{2}$$\sqrt{2}$ $\frac{2}{\sqrt{3}}$$\frac{2}{\sqrt{3}}$ $1$$1$ $\sec$$\sec$ $1$$1$ $\frac{2}{\sqrt{3}}$$\frac{2}{\sqrt{3}}$ $\sqrt{2}$$\sqrt{2}$ $2$$2$ Not Defined $\cot$$\cot$ Not Defined $\sqrt{3}$$\sqrt{3}$ $1$$1$ $\frac{1}{\sqrt{3}}$$\frac{1}{\sqrt{3}}$ $0$$0$

However, if we need to find the value of the trigonometrical ratios for angles that are other than the angles in the above tables, we use the trigonometric tables.

The trigonometrical tables give values of natural sines, cosines, and tangents to four decimal places. A trigonometrical table has three parts.

(i) a column on the left that has the degrees from $0 \ to \ 89$

(ii) ten columns on the top that has $0', 6', 12', 18', 24', 30', 36', 42', 48' \ and \ 54'$. These are minutes.

(iii) Five columns headed by $1', 2', 3', 4', \ and \ 5'$.

Note: $1^{\circ}$ is divided in $60$ minutes i.e. $1^{\circ} = 60'$

and $1'$ is divided into $60$ seconds i.e. $1' = 60''$