Trigonometric Identities

In mathematics, an identity is an equation or formula (often called an equality) that is always true for every possible value of its variable (or variables). A trigonometric identity is an identity that is written in terms of one or more trigonometric functions. There are a great number of trigonometric identities, many of which are only rarely used, so you should not attempt to memorise them. Just be aware that they exist, and perhaps familiarise yourself with the more commonly used ones, some of which are described below. For the identities shown, we have used the Greek letter alpha (α) to denote the angle if the identity relates to only a single angle. For identities that relate to two angles, we additionally use the Greek letter beta (β). Note that angles may be expressed in either degrees or radians. A value of pi (π) represents either pi radians or one hundred and eighty degrees (180°).

Although many problems involving trigonometry require the use of the sine or cosine functions, there are often times when other trigonometric functions are better suited to dealing with a particular situation. Knowing how the various functions relate to one another by being familiar with the trigonometric identities can be invaluable when it comes to choosing the most appropriate function (or functions) for the task at hand. Another advantage of trigonometric identities is that they allow us to replace an expression that uses one trigonometric function with an equivalent (and usually less complex) expression that uses a different trigonometric function. Being able to substitute one expression for another is often useful, especially when we need to simplify complex equations or formulas. Indeed, each of the six main trigonometric functions can be expressed in terms of any one of the other trigonometric functions. To see a table of trigonometric functions and their equivalents, click here.


The ratio identities

First let's look at two simple trigonometric identities usually referred to as the ratio identities. The term ratio identity could perhaps also be applied to other trigonometric identities, since all trigonometric functions can be defined as ratios. However, these are the only identities that describe trigonometric functions (namely the tangent and cotangent) as a simple ratio of sine and cosine.


tan (α)  =  sin (α)
cos (α)
cot (α)  =  cos (α)
sin (α)


The reciprocal identities

This is another set of basic trigonometric identities in which each of the six main trigonometric functions is described as a ratio. This time, the ratio in question is given as the reciprocal of another trigonometric function.


csc (α)  =  1
sin (α)
sin (α)  =  1
csc (α)
sec (α)  =  1
cos (α)
cos (α)  =  1
sec (α)
cot (α)  =  1
tan (α)
tan (α)  =  1
cot (α)


Trigonometric functions in terms of their complements

The following trigonometric identities express each of the main trigonometric functions in terms of their complements:

sin (α)  =  cos (π/2 - α)

cos (α)  =  sin (π/2 - α)

tan (α)  =  cot (π/2 - α)

csc (α)  =  sec (π/2 - α)

sec (α)  =  csc (π/2 - α)

cot (α)  =  tan (π/2 - α)


Trigonometric functions in terms of their supplements

The following trigonometric identities express each of the main trigonometric functions in terms of their supplements:

sin (α)  =  sin (π - α)

cos (α)  =  -cos (π - α)

tan (α)  =  -tan (π - α)

csc (α)  =  csc (π - α)

sec (α)  =  -sec (π - α)

cot (α)  =  -cot (π - α)


Periodicity of trigonometric functions

The sine, cosine, secant and cosecant trigonometric functions each have a period of three hundred and sixty degrees (360°) or . The tangent and cotangent functions each have a period of one hundred and eighty degrees (180°) or π. The following trigonometric identities simply express this periodicity:

sin (α)  =  sin (α + 2π)

cos (α)  =  -cos (α + 2π)

tan (α)  =  -tan (α + π)

csc (α)  =  csc (α + 2π)

sec (α)  =  -sec (α + 2π)

cot (α)  =  -cot (α + π)


Trigonometric identities for negative angles

All trigonometric functions can be described as either odd or even. A brief explanation (or reminder) might be in order here. Let's assume that we have a function ƒ(x). If the function is even, then for all real values of x, the following equation is always true:

ƒ(-x)  =  ƒ(x)

If the function is odd, then for all real values of x , the following equation is always true:

ƒ(-x)  =  -ƒ(x)

This particular characteristic of a trigonometric function (i.e. whether the function is odd or even) is called its parity. If you know whether a trigonometric function is odd or even it can sometimes help you to simplify a trigonometric expression, particularly if it contains a variable with a negative value. The sine, tangent, cosecant and cotangent functions are all odd functions, while the cosine and secant functions are even. The identities shown below are the trigonometric identities for negative angles, and reflect the parity (odd or even) of each function.

sin (-α)  =  -sin (α)

cos (-α)  =  cos (α)

tan (-α)  =  -tan (α)

csc (-α)  =  -csc (α)

sec (-α)  =  sec (α)

cot (-α)  =  -cot (α)


The Pythagorean identity

The Pythagorean identity is the trigonometric equivalent of Pythagoras' theorem in that it describes the relationship between the sides of a right angled-triangle. It derives from the equation x 2 + y 2 = 1 for any point on the perimeter of the unit circle, where x represents the cosine value and y represents the sine value.

cos2 (α)  +  sin2 (α)  =  1

The following identities are closely related to the Pythagorean identity. The first is the result of dividing the Pythagorean identity by sin2 (α):

cot2 (α)  +  1  =  csc2 (α)

The second is the result of dividing the Pythagorean identity by cos2 (α):

tan2 (α)  +  1  =  sec2 (α)


Angle sum and difference identities

We present below the angle sum and difference identities for the sine, cosine and tangent functions. Once you have had a chance to see what they look like, we will describe what they can be used for and provide some concrete examples.

The sum formulas:

sin (α + β)  =  sin (α· cos (β)  +  cos (α· sin (β)

cos (α + β)  =  cos (α· cos (β)  -  sin (α· sin (β)

tan (α + β)  =  tan (α)  +  tan (β)
1 - tan (α· tan (β)

The difference formulas:

sin (α - β)  =  sin (α· cos (β)  -  cos (α· sin (β)

cos (α - β)  =  cos (α· cos (β)  +  sin (α· sin (β)

tan (α - β)  =  tan (α)  -  tan (β)
tan (α)  -  tan (β)

The sum and difference identities, also known as the sum and difference formulas, can be used to find the sine, cosine or tangent of an angle that is the result of adding two angles together (the sum) or subtracting one angle from another (the difference). This can be very useful if we want to express the sine, cosine or tangent of an angle exactly, i.e. as a rational value. For example, supposing we want to find the trigonometric function value of an angle which measures one hundred and five degrees (105°) and express it exactly. Obviously, a calculator will give us a value, often to an accuracy of over thirty decimal places, but this is still an approximation. In many cases, an approximation is sufficient. Sometimes, however, we may wish to express the value exactly, perhaps for use in further calculations.

One hundred and five degrees is not one of those "nice" angles like sixty degrees or ninety degrees, for which we can look up the exact value of a trigonometric function (known as a trigonometric constant) in a table, or may even have committed to memory. Such angles are sometimes referred to as the primary solution angles, and consist of all the angles between zero and three hundred and sixty degrees that are multiples of either thirty degrees (30°) or forty-five degrees (45°). To see a table listing the primary solution angles and their trigonometric constants, click here. We can use the sum or difference of two such angles to get an angle of one hundred and five degrees. We could, for example, use any of the following expressions:

45° + 60°, 330° - 225°, 315° - 210°, 240° - 135°,

225° - 120°, 150° - 45°, 135° - 30°

Let's assume we want to find the sine of one hundred and five degrees. We'll use the sum of forty-five degrees and sixty degrees (45° + 60°) as an example. Here is the sum formula for the sine function:

sin (α + β)  =  sin (α· cos (β)  +  cos (α· sin (β)

Substituting actual values, we get:

sin (105)  =  sin (45) · cos (60)  +  cos (45) · sin (60)

Now we will substitute the corresponding trigonometric constants:

sin (105)  =  √2 · 1 + √2 · √3
2222
sin (105)  =  √2 + √6 = √2 + √6
444


Sum and product identities

The sum and product identities are often used when we are dealing with problems involving differential or integral calculus (frequently referred to as differentiation and integration respectively). They allow us to convert the product of two sine or cosine values into a sum, or vice versa. The identities shown below are collectively known as the sum identities, and are used to transform the sum or difference of the sine or cosine values of two angles into a product:

sin (α)  +  sin (β)  =  2 · sin (α + β) · cos (α - β)
22
sin (α)  -  sin (β)  =  2 · cos (α + β) · sin (α - β)
22
cos (α)  +  cos (β)  =  2 · cos (α + β) · cos (α - β)
22
cos (α)  -  cos (β)  =  2 · sin (α + β) · sin (α - β)
22

The following identities are collectively known as the product identities, and are used to transform the product of the sine or cosine values of two angles into a sum or difference:

sin (α· cos (β)  =  sin (α + β)  +  sin (α - β)
2
cos (α· sin (β)  =  sin (α + β)  -  sin (α - β)
2
cos (α· cos (β)  =  cos (α - β)  +  cos (α + β)
2
sin (α· sin (β)  =  cos (α - β)  -  cos (α + β)
2


Half-angle formulas

Half-angle formulas allow us to express the trigonometric function of an angle equal to α/2 in terms of α, which can often make life easier when it comes to performing complex calculations, such as those involving integration. The half-angle formulas are shown below.

sin (α)  =  ± √(1 - cos (α))
22
cos (α)  =  ± √(1 + cos (α))
22
tan (α)  =  ± √(1 - cos (α))  =  1 - cos (α) = sin (α)
21 + cos (α)sin (α)1 + cos (α)


Double-angle formulas

Double-angle formulas allow us to express the trigonometric function of an angle equal to 2α in terms of α. As with the half-angle formulas, this can often make life easier when it comes to performing complex calculations, such as those involving integration. The double-angle formulas are shown below.

sin (2α)  =  2 · sin (α· cos (α)

cos (2α)  =  cos2 (α)  -  sin2 (α)  =  2 · cos2 (α)  -  1  =  1 - 2 · sin2 (α)

tan (2α)  =  · tan (α)
1 - tan2 (α)


Power reduction formulas

Power reduction formulas can be derived using the half-angle and double-angle formulas, and the Pythagorean identity (cos2 (α) + sin2 (α) = 1). Like the half-angle and double-angle formulas, they allow us to simplify trigonometric expressions and can often make life easier when it comes to performing complex calculations. In this case, the formula takes an expression in which a trigonometric function is raised to a power, and transforms it into an expression in which the exponent is absent (hence "power reduction"). The power reduction formulas are shown below.

sin2 (α)  =  1 - cos (2α)
2
cos2 (α)  =  1 + cos (2α)
2
tan2 (α)  =  1 - cos (2α)
1 + cos (2α)