# What are the six trigonometry functions?

Trigonometry can be defined as the branch of mathematics that determines and studies the relationships between the sides of a triangle and angles subtended by them. Trigonometry is basically used in the case of right-angled triangles. Trigonometric functions define the relationships between the 3 sides and the angles of a triangle. There are 6 trigonometric functions mainly. Before going into the study of the trigonometric functions we will learn about the 3 sides of a right-angled triangle.

The three sides of a right-angled triangle are as follows,

**Base**The side on which the angle Î¸ lies is known as the base.**Perpendicular**It is the side opposite to the angle Î¸ in consideration.**Hypotenuse**It is the longest side in a right-angled triangle and opposite to the 90Â° angle.

### Trigonometric Functions

Trigonometry has 6 basic trigonometric functions, they are sine, cosine, tangent, cosecant, secant, and cotangent. Now let’s look into the trigonometric functions. The six trigonometric functions are as follows,

**sine**It is represented as sin Î¸ and is defined as the ratio of perpendicular and hypotenuse.**cosine**It is represented as cos Î¸ and is defined as the ratio of base and hypotenuse.**tangent**It is represented as tan Î¸ and is defined as the ratio of sine and cosine of an angle. Thus the definition of tangent comes out to be the ratio of perpendicular and base.**cosecant**It is the reciprocal of sin Î¸ and is represented as cosec Î¸.**secant**It is the reciprocal of cos Î¸ and is represented as sec Î¸.**cotangent**It is the reciprocal of tan Î¸ and is represented as cot Î¸.

### What are the six trigonometry functions?

The six trigonometric functions have formulae for the right-angled triangles, the formulae help in identifying the lengths of the sides of a right-angled triangle, lets take a look at all those formulae,

Trigonometric Functions |
Formulae |

sin Î¸ | |

cos Î¸ | |

tan Î¸ | |

cosec Î¸ | |

sec Î¸ | |

cot Î¸ |

The below table shows the values of these functions at some standard angles,

Function |
0Â° |
30Â° |
45Â° |
60Â° |
90Â° |

âˆž | |||||

âˆž | |||||

âˆž | |||||

âˆž |

Note:It is advised to remember the first 3 trigonometric functions and their values at these standard angles for ease of calculations.

### Sample Problems

**Question 1: Evaluate sine, cosine, and tangent in the following figure**.

**Solution:**

Given

Using the trigonometric formulas for sine, cosine and tangent,

**Question 2: In the same triangle evaluate secant, cosecant**,** and cotangent. **

**Solution: **

As it is known the values of sine, cosine and tangent, we can easily calculate the required ratios.

**Question 3: Given ****, evaluate **sin Î¸.cos Î¸**.**

**Solution: **

Thus P=6, B=8

Using Pythagoras theorem,

H

^{2}=P^{2}+B^{2}H

^{2}=36+64=100Therefore, H =10

Now,

**Question 4: If ****, evaluate **tan^{2}Î¸.

**Solution: **

Given

Thus

**Question 5: In the given triangle, verify** sin^{2}Î¸+cos^{2}Î¸ = 1

**Solution: **

Given P=12, B=5, H=13

Thus

Hence verified.

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