This particular characteristic of a trigonometric function (i.e. If the function is odd, then for all real values of x, the following equation is always true: If the function is even, then for all real values of x, the following equation is always true: Let's assume that we have a function ƒ( x). A brief explanation (or reminder) might be in order here. To see a table of trigonometric functions and their equivalents, click here.Īll trigonometric functions can be described as either odd or even. Indeed, each of the six main trigonometric functions can be expressed in terms of any one of the other trigonometric functions. Being able to substitute one expression for another is often useful, especially when we need to simplify complex equations or formulas. 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.
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. A value of pi ( π) represents either pi radians or one hundred and eighty degrees ( 180°).Īlthough 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. Note that angles may be expressed in either degrees or radians. For identities that relate to two angles, we additionally use the Greek letter beta ( β). 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. Just be aware that they exist, and perhaps familiarise yourself with the more commonly used ones, some of which are described below. There are a great number of trigonometric identities, many of which are only rarely used, so you should not attempt to memorise them. A trigonometric identity is an identity that is written in terms of one or more trigonometric functions. The following have equivalent value, and we can use whichever one we like, depending on the situation:įind cos 60° by using the functions of 30°.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). = 2cos 2 α − 1 Summary - Cosine of a Double Angle Likewise, we can substitute (1 − cos 2 α) for sin 2 α into our RHS and obtain: RHS = cos α cos α − sin α sin α = cos 2 α − sin 2 α Different forms of the Cosine Double Angle Resultīy using the result sin 2 α + cos 2 α = 1, (which we found in Trigonometric Identities) we can write the RHS of the above formula as: This time we start with the cosine of the sum of two angles:Īnd once again replace β with α on both the LHS and RHS, as follows: Using a similar process, we obtain the cosine of a double angle formula: It is useful for simplifying expressions later. This result is called the sine of a double angle. Putting our results for the LHS and RHS together, we obtain the important result:
Sin α cos α + cos α sin α = 2 sin α cos α
Since we replaced β in the LHS with α, we need to do the same on the right side. We will use this to obtain the sine of a double angle. Recall from the last section, the sine of the sum of two angles: In this way, you will understand it better and have less to clutter your memory with. With these formulas, it is better to remember where they come from, rather than trying to remember the actual formulas. The double-angle formulas can be quite useful when we need to simplify complicated trigonometric expressions later.
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