Trigonometry is composed of two words: *Trigon* and *Metron*. Trigon means ‘triangle’ and metron means ‘to measure’. Combined, it means measuring sides or angles of a triangle and that is what trigonometry precisely is. It is the study of angles of triangles and relationships between them. **Trigonometric Functions **can be defined as ‘The functions which help us relate the angles and sides of triangles.’

Trigonometric functions are also known as **circular functions**. This is because they are normally explained and derived using **unit circles** (circles with radius equal to 1 unit). Let us define these functions now.

**Trigonometric Ratios:**

In earlier grades, we have seen the trigonometric ratios for acute angles (angles less than \(90^{\circ}\)

In figure 1, \(\triangle ABC\)

\(\sin~\theta\)

\(\cos~\theta\)

\(\tan~\theta\)

Let us take a unit circle as shown in the figure 2. The radius of the circle is 1 unit. The length of the arc AP is \(x\)

- \(sin~x\)
= \(\large\frac{PM}{OP}\) = \(\large \frac{n}{1}\) = \(n\) - \(cos~x\)
= \(\large \frac{OM}{OP}\) = \(\large\frac{m}{1}\) = \(m\)

\(\triangle OMP\)

\(OM^2 + MP^2\)

\(m^2 + n^2\)

From the results we obtained before, we get

\(\cos^2~x + \sin^2~x\)

Dividing both the sides by \(\cos^2~x\)

\(1 + \tan^2~x\)

Dividing both sides now by \(tan^2~x\)

\(1 + cot^2~x\)

The identities written above are true for any point taken on the circle. The circumference of the circle is equal to 2π units. Also, the full angle is equal to 2π radians. So,

\(∠AOB\)

\(∠AOD\)

If a given angle is an integral multiple of \(\large \frac{π}{2}\)

Yes, it is. This is the reason due to which these functions are said to be periodic. Even after n revolutions, at P, value of sin x will still be equal to m. This means

\(\sin~(nπ+x)\)

There are similar conclusions that can be drawn for other ratios as well. This also means that the equation \(\sin~x\)

\(x\)

Similarly, if \(\cos~x\)

\(x\)

Table 1: Values of different trigonometric ratios

The following table gives the value of all the trigonometric ratios for common angles.

Angles (In Degrees) |
0° |
30° |
45° |
60° |
90° |
180° |
270° |
360° |

Angles (In Radians) |
0 |
\(\frac{π}{6}\) |
\(\frac{π}{4}\) |
\(\frac{π}{3}\) |
\(\frac{π}{2}\) |
π |
\(\frac{3π}{2}\) |
2π |

sin | 0 | \(\frac{1}{2}\) |
\(\frac{1}{√2}\) |
\(\frac{√3}{2}\) |
1 | 0 | -1 | 0 |

cos | 1 | \(\frac{√3}{2}\) |
\(\frac{1}{√2}\) |
\(\frac{1}{2}\) |
0 | -1 | 0 | 1 |

tan | 0 | \(\frac{1}{√3}\) |
1 | √3 | N D | 0 | N D | 0 |

cot | N D | √3 | 1 | \(\frac{1}{√3}\) |
0 | N D | 0 | N D |

csc | N D | 2 | √2 | \(\frac{2}{√3}\) |
1 | N D | -1 | N D |

sec | 1 | \(\frac{2}{√3}\) |
√2 | 2 | N D | -1 | N D | 1 |

To learn how to decide signs of trigonometric functions in different quadrants and how to calculate their value, download BYJU’S-The Learning App and experience the fun in learning.

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