Trigonometric Functions

# Trigonometric Functions

## Six Trigonometric Functions Chart

The six trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent. They are all ratios of two lengths, typically angles, and they are all periodic. The period of a trigonometric function is the length of time it takes for the function to repeat its cycle.

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The sine, cosine, and tangent functions are usually abbreviated sin, cos, and tan. The cosecant, secant, and cotangent functions are abbreviated csc, sec, and cot.

The following chart shows the six trigonometric functions and their period.

### Sine Function

Sine Function of an angle is the ratio between the opposite side length to that of the hypotenuse. From the above diagram, the value of sin will be:
• Sin a =Opposite/Hypotenuse = CB/CA

### Cos Function

Cos of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. From the above diagram, the cos function will be derived as follows.

• Cos a = Adjacent/Hypotenuse = AB/CA

### Tan Function

The tangent function is the ratio of the length of the opposite side to that of the adjacent side. It should be noted that the tan can also be represented in terms of sine and cos as their ratio. From the diagram taken above, the tan function will be the following.

• Tan a = Opposite/Adjacent = CB/BA

Also, in terms of sine and cos, tan can be represented as:

Tan a = sin a/cos a

### Secant, Cosecant and Cotangent Functions

Secant, cosecant (csc) and cotangent are the three additional functions which are derived from the primary functions of sine, cos, and tan. The reciprocal of sine, cos, and tan are cosecant (csc), secant (sec), and cotangent (cot) respectively. The formula of each of these functions are given as:

• Sec a = 1/(cos a) = Hypotenuse/Adjacent = CA/AB
• Cosec a = 1/(sin a) = Hypotenuse/Opposite = CA/CB
• cot a = 1/(tan a) = Adjacent/Opposite = BA/CB

Note: inverse Trigonometric Functions are used to obtain an angle from any of the angle’s trigonometric ratios. Basically, inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions are represented as arcsine, arccosine, arctangent, arc cotangent, arc secant, and arc cosecant

## Formulas

Let us discuss the formulas given in the table below for functions of trigonometric ratios(sine, cosine, tangent, cotangent, secant and cosecant) for a right-angled triangle.

 Formulas for Angle θ Reciprocal Identities sin θ = Opposite Side/Hypotenuse sin θ = 1/cosec θ cos θ = Adjacent Side/Hypotenuse cos θ = 1/sec θ tan θ = Opposite Side/Adjacent tan θ = 1/cot θ cot θ = Adjacent Side/Opposite cot θ = 1/tan θ sec θ = Hypotenuse/Adjacent Side sec θ = 1/cos θ cosec θ = Hypotenuse/Opposite cosec θ = 1/sin θ

## Identities

Below are the identities related to trig functions.

### Even and Odd functions

The cos and sec functions are even functions; the rest other functions are odd functions.

sin(-x) = -sin x

cos(-x) = cos x

tan(-x) = – tan x

cot(-x) = -cot x

csc(-x) = -csc x

sec(-x) = sec x

### Periodic Functions

The trig functions are the periodic functions. The smallest periodic cycle is 2π but for tangent and the cotangent it is π.

sin(x+2nπ) = sin x

cos(x+2nπ) = cos x

tan(x+nπ) = tan x

cot(x+nπ) = cot x

csc(x+2nπ) = csc x

sec(x+2nπ) = sec x

Where n is any integer.

### Pythagorean Identities

When the Pythagoras theorem is expressed in the form of trigonometry functions, it is said to be Pythagorean identity. There are majorly three identities:

• sin2 x + cos2 x = 1 [Very Important]
• 1+tan2 x = sec2 x
• cosec2 x = 1 + cot2 x

These three identities are of great importance in Mathematics, as most of the trigonometry questions are prepared in exams based on them. Therefore, students should memorise these identities to solve such problems easily.

### Sum and Difference Identities

• sin(x+y) = sin(x).cos(y)+cos(x).sin(y)
• sin(x–y) = sin(x).cos(y)–cos(x).sin(y)
• cos(x+y) = cosx.cosy–sinx.siny
• cos(x–y) = cosx.cosy+sinx.siny
• tan(x+y) = [tan(x)+tan(y)]/[1-tan(x)tan(y)]
• tan(x-y) = [tan(x)-tan(y)]/[1+tan(x)tan(y)]

For more visit Trigonometric Ratios of Standard Angles

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