# Trigonometric Functions

## What are Trigonometric Functions?

In mathematics, there are a total of six trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant that symbolize ratios of sides of right triangles. They are also called circular functions because their values can be simply described as ratios of the x and y coordinates of points on a circle of radius 1 that correspond to angles in standard positions.

Note that in these modern times, such values have been tabulated and programmed into scientific calculators and computers. It has immensely helped trigonometry to get easily used for surveying, engineering, and navigation problems, wherein the length of a side along with one of a right triangle’s acute angles are known and the lengths of the other sides are to be calculated. The primary trigonometric identity can be written as sin2θ + cos2θ = 1, in which θ is an angle. Some intrinsic qualities of the trigonometric functions make them extremely useful in mathematical analysis. Especially, their derivatives create patterns that are useful for solving differential equations.

**Domain and Range of Trigonometric Ratios**

Trigonometric Function |
Domain |
Range |

sin x | R | -1 ≤ sin x ≤ 1 |

cos x | R | -1 ≤ cos x ≤ 1 |

tan x | R – {(2n + 1)π/2, n ∈ I} | R |

cosec x | R – {nπ, n ∈ I} | R – {x: -1 < x < 1} |

sec x | R – {(2n + 1)π/2, n ∈ I} | R – {x: -1 < x < 1} |

cot x | R – {nπ, n ∈ I} | R |

** **These functions are useful for relating the angles of a triangle with the sides of that triangle. Trigonometric functions are quite crucial while studying triangles and modeling periodic phenomena such as light, waves and sound.

For defining these functions for the angle *theta,* let’s start with a right triangle. Every function relates the angle to two sides of a right triangle. So, let’s define the sides of the triangle.

- The
**hypotenuse**is the side opposite the right angle. The hypotenuse is always the longest side of a right triangle. - The
**opposite**side is the side opposite to the angle we are interested in,*theta.* - The
**adjacent**side is the side that has both the angles of interest (angle*theta*and the right angle).

Here is the relationship between the trigonometric functions and the sides of the triangle:

- sine(
*theta*) = opposite / hypotenuse - cosecant(
*theta*) = hypotenuse / opposite - cosine(
*theta*) = adjacent / hypotenuse - secant(
*theta*) = hypotenuse / adjacent - tangent(
*theta*) = opposite / adjacent - cotangent(
*theta*) = adjacent / opposite

**Take a look at the trigonometry table of the basic ratios for a few frequently used angles. These formulae need to be memorised as they are used in most of the questions:**

Angle (x) |
sin x |
cos x |
tan x |
cosec x |
sec x |
cot x |

0° | 0 | 1 | 0 | undefined | 1 | undefined |

90° = π/2 | 1 | 0 | undefined | 1 | undefined | 0 |

180° = π | 0 | -1 | 0 | undefined | -1 | undefined |

270° =3π/2 | -1 | 0 | undefined | -1 | undefined | 0 |

360° = 2π | 0 | 1 | 0 | undefined | 1 | undefined |

**More Formulae of Trigonometry**

Angles |
0° |
30° |
45° |
60° |
90° |

sin |
0 | 1/2 | 1/√2 | √3/2 | 1 |

cos |
1 | √3/2 | 1/√2 | 1/2 | 0 |

tan |
0 | √3/2 | 1 | √3 | undefined |

cosec |
undefined | 2 | √2 | 2/√3 | 1 |

sec |
1 | 2/√3 | √2 | 2 | undefined |

cot |
undefined | √3 | 1 |

**Periods of Various Trigonometric Functions**

1) sin x has period 2π

2) cos x has period 2π

3) tan x has period π

4) sin(ax + b), cos (ax + b), sec(ax + b), cosec (ax + b) all are of period 2π/a

5) tan (ax + b) and cot (ax + b) have π/a as their period

6) |sin (ax + b)|, |cos (ax + b)|, |sec(ax + b)|, |cosec (ax + b)| all are of period π/a

7) |tan (ax + b)| and |cot (ax + b)| have π/2a as their period

**Sum and Difference Formulae of Trigonometric Ratios**

1) sin (a + ß) = sin(a) cos(ß) + cos(a) sin(ß)

2) sin (a – ß) = sin(a) cos(ß) – cos(a) sin(ß)

3) cos (a + ß) = cos(a) cos(ß) – sin(a) sin(ß)

4) cos (a – ß) = cos(a) cos(ß) + sin(a) sin(ß)

5) tan (a + ß) = [tan(a) + tan (ß)]/ [1 – tan(a) tan(ß)]

6) tan (a – ß) = [tan(a) – tan (ß)]/ [1 + tan (a) tan (ß)]

7) tan (π/4 + θ) = (1 + tan θ)/(1 – tan θ)

8) tan (π/4 – θ) = (1 – tan θ)/(1 + tan θ)

9) cot (a + ß) = [cot(a) . cot (ß) – 1]/ [cot (a) + cot (ß)]

10) cot (a – ß) = [cot(a) . cot (ß) + 1]/ [cot (ß) – cot (a)]

**Double or Triple -Angle Identities**

1) sin 2x = 2sin x cos x

2) cos2x = cos^{2}x – sin^{2}x = 1 – 2sin^{2}x = 2cos^{2}x – 1

3) tan 2x = 2 tan x / (1-tan ^{2}x)

4) sin 3x = 3 sin x – 4 sin^{3}x

5) cos3x = 4 cos^{3}x – 3 cosx

6) tan 3x = (3 tan x – tan^{3}x) / (1- 3tan ^{2}x)

**For Angles A, B and C**

1) sin (A + B + C) = sin A cos B cos C + cos A sin B cos C + cos A cos B sin C – sin A sin B sin C

2) cos (A + B + C) = cos A cos B cos C – cos A sin B sin C – sin A cos B sin C – sin A sin B cos C

3) tan (A + B + C) = [tan A + tan B + tan C – tan A tan B tan C]/ [1 – tan A tan B – tan B tan C – tan A tan C

4) cot (A + B + C) = [cot A cot B cot C – cot A – cot B – cot C]/ [cot A cot B + cot B cot C + cot A cot C – 1]

**How to Solve a Trigonometric Equation**

1) Always try to reduce the equation in terms of any one variable, preferably x. After that, tackle the equation just the way you would in case of a single variable.

2) It’s a good idea to derive the linear/algebraic simultaneous equations from the given trigonometric equations and then solve them as algebraic simultaneous equations.

3) Many times, you might need to make substitutions. It is ideally helpful when the system includes only two trigonometric functions.

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