The word Trigonometry can be split as Tri - gono - metry, meaning the measure of the three angles of a triangle. The study of trigonometry is very much useful in finding the distance between the stars and planets, height of a building, distance of a ship from the shore, height of a plane at an instant etc. To solve the problems on this we should know the basics of trigonometry. To study about the basics of trigonometry, let us study about the right angled triangle and the trigonometric ratios of each of the acute angles.

Sine, Cosine and Tangents are the ratio of the sides of a right angled triangle. In a right angled triangle, let us consider one of the right angled triangle and denote it as $\theta$
Sine Cosine Tangent Formulas
Therefore, Sine of the angle
$\theta$ is denoted as sin$\theta$ which is the ratio of the opposite side to the hypotenuse.
which is denoted as
sin $\theta$ = $\frac{Opposite\; side}{Hypotenuse}$
= $\frac{AB}{AC}$

Cosine of the angle $\theta$ is denoted as cos$\theta$ which is the ratio of the adjacent side to the hypotenuse of a triangle.
cos $\theta$ = $\frac{Adjacent\;side}{Hypotenuse}$
= $\frac{CB}{AC}$

Tangent of the angle $\theta$ is denoted as tan$\theta$ which is the ratio of the opposite side to the adjacent side of the triangle.
tan $\theta$ = $\frac{Opposite\; side}{Adjacent\; Side}$

= $\frac{AB}{BC}$


Sine Cosine and Tangent Ratios:


Let us consider the following diagram containing showing the perpendicular and base of the triangle.

Sine Cosine and Tangent

The base of the triangle is b unites, the perpendicular side to the base is a units.
Therefore, According to Pythagoras theorem, the length of the hypotenuse c is given by $\sqrt{(a^{2}+b^{2}}$
sin $\theta$ = $\frac{Perpendicular}{Hypotenuse}$

= $\frac{a}{c}$

cos
$\theta$ = $\frac{Base}{Hypotenuse}$

= $\frac{b}{c}$


tan
$\theta$ = $\frac{Perpendicular}{Base}$

= $\frac{a}{b}$

Solved Example

Question: Find the three trigonometric ratios of the angle A of the triangle whose base measure 5 cm and the Perpendicular measure 12 cm.
Solution:
 
The following diagram shows the angle A for which the trigonometric ratios are to be written.

Sine Cosine Tangent Example

We can find the length of the hypotenuse using the formula,
                                           c =
$\sqrt{(a^{2}+b^{2})}$

                                              = $\sqrt{(12^{2}+5^{2})}$


                                              = $\sqrt{(144 + 25)}$


                                              = $\sqrt{169}$

                                              = 13
Therefore,
                            sin $\theta$ =  $\frac{perpendicular}{Hypotenuse}$

                                      =
$\frac{12}{13}$

                           cos $\theta$$\frac{Base}{Hypotenuse}$

                                      =
$\frac{5}{13}$

                            tan
$\theta$ = $\frac{Perpendicular}{Base}$

                                      =
$\frac{12}{5}$