Adding and Subtracting: Trigonometric Identities
Adding and Subtracting: Trigonometric Identities refer to key mathematical principles that facilitate the manipulation of trigonometric functions through operations of addition and subtraction. These identities are essential for solving trigonometric equations and simplifying expressions, particularly in calculus. They can be categorized into three main types: sum and difference identities, sum-to-product identities, and product-to-sum identities.
Sum and difference identities arise from the relationships between angles, enabling the derivation of trigonometric function values for angles not explicitly defined on the unit circle. Sum-to-product identities and product-to-sum identities help in transforming trigonometric expressions, which can simplify integration tasks and solve complex equations. For example, these identities can convert challenging functions into more manageable forms, making them easier to integrate or factor. Overall, a solid understanding of these identities is beneficial for students and professionals working with trigonometric functions in various applications.
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Adding and Subtracting: Trigonometric Identities
Trigonometric identities are important for solving trigonometric equations and simplifying expressions that arise in calculus. This article focuses on trigonometric identities that involve addition and subtraction.
An identity in a variable
on a set
is an equation that holds true for all values of
. For example,
.
.
The third example above is a trigonometric identity, which is part of analytic trigonometry. Addition and subtraction operations arise in trigonometric identities in three important ways: 1) addition and subtraction of angles, as the functions 
and 
; 2) addition and subtraction of trigonometric functions, as in 
and 
; and combinations of both of the above.
The identities involving the first type of expression are called sum and difference identities, the identities involving the second type of expression are called sum-to-product identities, and identities involving combinations of both types are called product-to-sum identities. Familiarity with the following identities will prove helpful for reading this article.
Overview
The sum and difference identities can be derived from the unit circle. See Figures 1a and 1b.
The distance d is the same in both figures. Writing the distance formula for each figure yields the following.
Equating (6) and (7) and solving yields
Letting
in (8) and applying (4) and (5) yields
Replacing u with
in (8) and applying (2) and (3) yields
Letting
in (10) and applying (4) and (5) yields
The sum and difference identities for tangents can be obtained by using (8), (9), (10), and (11) with (1).
One useful application of identities (8) – (13) is finding exact values of trigonometric functions of angles that are not listed on the typical unit circle. For instance
and
are not listed on the unit circle, but the exact values of all six trigonometric functions can nevertheless be found using (8) – (13). This is because
and
are expressible as
and
, respectively, and the angles
and
do appear on the typical unit circle. The following two examples should suffice to show this point.
The next set of identities to be discussed here are four product-to-sum identities. Alternately adding and subtracting (8) and (9) to solve for
and
, respectively, yields the following.
Next, alternately adding and subtracting (10) and (11) to solve for
and
, respectively, yields the following.
Identities (14) – (17) are particularly useful for integration in calculus. For example,
is a difficult function to integrate. Applying (16) to the function yields
, which is an easy function to integrate.
The final set of identities to be discussed in this article are four sum-to-product identities. Replacing
with
and
with
in
yields the following.
One useful application of these identities is in solving trigonometric equations. For example the equation
is difficult to solve as is. But (21) can be applied to the left side to transform the equation into
. This can now be solved by factoring as follows.
Bibliography
Larson, Ron, and Bruce H. Edwards. Calculus. Boston: Cengage, 2014. Print.
Larson, Ron, and David C. Falvo. Precalculus. Boston: Cengage, 2014. Print.
McKeague, Charles P, and Mark D. Turner. Trigonometry. Australia: Cengage, 2013. Print.
Stewart, James, L. Redlin, and Saleem Watson. Precalculus : Mathematics for Calculus. Belmont: Cengage, 2012. Print.
Sullivan, Michael. Precalculus : Enhanced with Graphing Utilities. Upper Saddle River: Pearson, 2013. Print.
Young, Cynthia Y. College Algebra. Hoboken: Wiley, 2012. Print.