Trigonometric identities can be very useful in simplifying seemingly complex trigonometric expressions and equations into much more manageable forms. They may seem daunting and imposing at first, but they really boil down to just a few simple formulas. In this guide, you’ll learn these identities along with clear and easy-to-follow examples of their practical use.
Trigonometric identities are equalities that involve trigonometric functions that are true for every single value of the occurring variables. To put that in other words, they are equations that hold true regardless of the values of the angles being chosen.
Trigonometric identities serve a purpose of not only defining the groundwork for trigonometry as a subject, but also for helping one simplify and solve what would otherwise be challenging problems. By utilizing identities, a complex trigonometric equation can be converted into a form that’s easier to digest via standard algebra tools.
Reciprocal trigonometric identities involve the relation between the core six trigonometric functions: sine, cosine, tangent, secant, cosecant, and the cotangent. These identities express each function via a formula that makes it equal to one over another trig function.
The six identities are:
Thus, the reciprocal identities can simplify equations that involve multiple trigonometric functions into ones consisting of only one function type. For example, the expression can be converted into a much simpler formula. Since , we can rewrite the expression as which can be further simplified to . This can in turn be further reduced to via the rules governing the addition of fractions. We could have also alternatively used the identity and obtained a result in terms of the cosecant, which would be .
Pythagorean trigonometric identities, as their name suggests, were discovered by the Greek mathematician Pythagoras and also represent equations that hold true regardless of the angle in question. The three core identities are:
These identities can also be utilized to help simplify trigonometric expressions and equations. For example, let’s utilize the first Pythagorean identity so simplify the expression . If we break apart the first term, we can see that the rest of the expression holds the trigonometric identity: . Working on it further, we obtain which at the end becomes just since the sum of the sine squared and cosine squared equals one.
Utilizing the second and third identities yields similar benefits. For example, we can utilize the second Pythagorean identity in combination with the reciprocal identity from the first section of this guide to simplify into and further into which is equal to 1.
Ratio identities represent relations in the ratios of core trigonometric functions, namely the sine and the cosine. They are alternatively labeled quotient trigonometric identities since the quotient of the sine and the cosine functions is utilized to derive the functions for the tangent and the cotangent. Quite simply, the identities are:
Let’s use these identities to once again simplify complex trigonometric expression into much more manageable ones. For example, consider the daunting-looking . We can use the ratio identities to break down the cotangent and the tangent terms into their sine and cosine components. In applying the identities to the above expression, we obtain . The expression looks even “bigger” now, but the path to simplicity is much more clear—we simply apply the rules of division and reduce the terms in the fraction. On the left side, the cosine squares cancel each other out and on the right side, the sin squared terms also reduce down to one. Thus, we obtain a much simpler expression of which, as we know already, equals to just one. Quite a big difference between that and the original scary-looking expression, isn’t it?
Double angle identities take a departure from the aforementioned trigonometric identities in the sense that they help deal with functions where the argument is not just Θ but 2Θ or other even multiples of it. Thus, an expression such involving elements such as can be broken down into simpler components we are used to working with before: functions of the sine, cosine, and tangent.
Here are the three main double angle identities:
, which can be alternatively expressed as or
As with the previous identities, let’s see it an action and apply the formula to help us simplify trigonometric expressions. Suppose we want to reduce . We can apply the trigonometric identity for alternatively expressing and obtain which conveniently further reduces to .
Make sure to not confuse and , they are very different expressions! For example, if we take Θ to be equal 45°, then equals , which is equal to 1 versus which equals or, alternatively, ~1.414.
Let’s try another example with the double angle formula for cosine. Suppose we try to simplify . If we take one of the forms of the double angle formula for the cosine, the term can be expressed as resulting in our original expression being equal to . In simplifying the numerator, we obtain which further reduces to once we apply the ratio identity for the cotangent.
With a little practice, trigonometric identities can become trusted members of your mathematical “toolbox.” And remember, if one approach to simplification does not give you the result you’re looking for, keep trying with other trigonometric identities; eventually a skilled application of the needed formula will produce the simple expression needed for the solution. Happy simplifying!
by Izolda Fotiyeva, Ph.D., and Dmitriy Fotiyeva, authors of The Complete Idiot’s Guide to Trigonometry