Tangent is therefore an odd function, which means that tan ( − θ ) = − tan ( θ ) tan ( − θ ) = − tan ( θ ) for all θ θ in the domain of the tangent function. We can interpret the tangent of a negative angle as tan (− θ ) = sin ( − θ ) cos (− θ ) = − sin θ cos θ = − tan θ. For example, consider the tangent identity, tan (− θ ) = −tan θ. The other even-odd identities follow from the even and odd nature of the sine and cosine functions. Since, cos (− θ ) = cos θ, cos (− θ ) = cos θ, cosine is an even function.Since sin (− θ ) = − sin θ, sin (− θ ) = − sin θ, sine is an odd function.We have already seen and used the first of these identifies, but now we will also use additional identities.įor all θ θ in the domain of the sine and cosine functions, respectively, we can state the following: We will begin with the Pythagorean Identities (see Table 1), which are equations involving trigonometric functions based on the properties of a right triangle. In this first section, we will work with the fundamental identities: the Pythagorean Identities, the even-odd identities, the reciprocal identities, and the quotient identities. Sometimes we have to factor expressions, expand expressions, find common denominators, or use other algebraic strategies to obtain the desired result. To verify the trigonometric identities, we usually start with the more complicated side of the equation and essentially rewrite the expression until it has been transformed into the same expression as the other side of the equation. Consequently, any trigonometric identity can be written in many ways. We already know that all of the trigonometric functions are related because they all are defined in terms of the unit circle. Basic properties and formulas of algebra, such as the difference of squares formula and the perfect squares formula, will simplify the work involved with trigonometric expressions and equations. In fact, we use algebraic techniques constantly to simplify trigonometric expressions. They are the basic tools of trigonometry used in solving trigonometric equations, just as factoring, finding common denominators, and using special formulas are the basic tools of solving algebraic equations. Identities enable us to simplify complicated expressions. Verifying the Fundamental Trigonometric Identities In this section, we will begin an examination of the fundamental trigonometric identities, including how we can verify them and how we can use them to simplify trigonometric expressions. Just as a spy will choose an Italian passport when traveling to Italy, we choose the identity that applies to the given scenario when solving a trigonometric equation. The trigonometric identities act in a similar manner to multiple passports-there are many ways to represent the same trigonometric expression. However, we know that each of those passports represents the same person. In espionage movies, we see international spies with multiple passports, each claiming a different identity. Note: Maple does not rationalize the denominator, that is, write the expression in the form A B sin x + C D for polynomials A, B, C, and D in cos x because this form usually leads to a result that is larger in total degree.Figure 1 International passports and travel documents In particular, any common factor between N and D has been cancelled out. If the input is a rational expression in sin x and cos x then an algorithm is used to put it in the form N D and reduce N and D to lowest terms such that the total degree of the numerator N (in sin x and cos x ) plus the total degree of the denominator is minimized. If the input involves multiple angles that are integer multiples of each other, for example, sin x, sin 2 x, and cos x 2 then the trigonometric functions are expressed in terms of a common angle, in this case x 2. Simplify expr, sin x 2 + cos x 2 − 1, sin x To apply the identity to reduce the polynomial so that the degree in sin x is at most 1, use the command If the input is a polynomial in sinh x and cosh x then simplify/trig yields a similar result using the identity cosh x 2 − sinh x 2 = 1. Where A and B are polynomials in cos x. If the input is a polynomial in sin x and cos x then simplify/trig factors out powers of sin x and cos x and applies the identity sin x 2 + cos x 2 = 1 to what is left so that the degree of what is left in sin x is at most 1. The simplify(expr,trig) calling sequence simplifies trigonometric expressions by applying the trigonometric identities sin x 2 + cos x 2 = 1 and cosh x 2 − sinh x 2 = 1.
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