Identifying Algebraic Properties Most Often Used When Solving Identities
Solving identities is almost a rite of passage for those studying trigonometry. Tackling the prospect of solving identities — and later simplifying trig expressions in calculus — goes much more smoothly if you have some algebraic tools at hand. With a plan of action, you’ll succeed more quickly and efficiently and have the desired product.
When solving an identity, you do bring in some trig substitutions (basic identities such as sin2 x + cos2 x = 1), but all your work has its main basis in algebraic rules and techniques. Here are the algebraic properties most commonly found when working with identities:
Commutative property of addition and multiplication: 2 sin x + sin y + sin x = 2 sin x + sin x + sin y and
You can change the order of the terms or factors to make combining terms more convenient.
Associative property of addition and multiplication: 2 sin x + (sin x + sin y) = (2 sin x + sin x) and
By regrouping terms or factors, you can add or multiply terms that combine.
Distributive property of multiplication over addition: sin x(1 – csc x) = sin x – sin x csc x. The distributive property is very useful, especially when you recognize that one of the products turns out to be a function times its reciprocal.
Doing a flip-flop of the two sides can make for more convenience in the work or when solving an equation.
Multiplication property of equations: If
then 2 sin x = 1. You can multiply both sides of an equation by the same number (just not 0). When solving a trig equation, you have many hidden opportunities to multiply each side of an equation by 0 or divide (multiply by a reciprocal) by 0. The trig functions sine, cosine, tangent, and cotangent are 0 for many angle measures. Just take those angle measures into account when determining a solution to the equation (in other words, throw them out).
Squaring a binomial: (sin x + cos x)2 = sin2 x + 2 sin x cos x + cos2 x. One of the most frequent errors found when squaring a binomial is forgetting that middle term. Squaring binomials is especially useful in trigonometry, because it tends to create terms that are a part of one of the Pythagorean identities.
Factoring (greatest common factor): sin2 x tan2 x – tan2 x = tan2 x(sin2 x – 1). When two or more terms have a common factor, dividing each term by that factor creates one or more workable expressions. Just be sure to divide all terms by the factor and to preserve the correct signs. When dividing by a negative factor, the signs all switch.
Factoring (difference of squares): sec2 x – 1 = (sec x – 1)(sec x + 1). The Pythagorean identities all have three squared terms in their equations. This lends to many opportunities to factor as the difference of squares. You look ahead to see what may then be divided out in a future step. Other factoring techniques are used less frequently, but don’t hesitate to refer back to your algebra to dredge up something not mentioned here.