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Pre-Calculus 12A Section 6.1, 6.3, 6.4 Reciprocal, Quotient, and Pythagorean Identities In mathematics, it is important to understand the difference between an equation and an identity. An equation is only true for certain value(s) of the variable. For example, the equation x 2 4 0 is true only for x 2 . An identity is an equation that is true for all values of the variable. For example: the equation x 2 2 x 2 4x 4 is true for all values of x, so we can call it an identity. A trigonometric identity is a trigonometric equation that is true for all permissible values of the variable. An example of a trigonometric equation is sin x = 1, which is true only for x 90 360n,n I An example of a trigonometric identity is sin2 x cos2 x 1 , which is true for all values of x. An example of a trigonometric identity with non-permissible values is csc values of x except 0 180n,n I , which is where sin 0 . 1 , which is true for all sin You are already familiar with the reciprocal and quotient identities. RECIPROCAL IDENTITIES sin 1 csc cos 1 sec tan 1 cot csc 1 sin sec 1 cos cot 1 tan QUOTIENT IDENTITIES tan sin cos cot cos sin These identities will be used to prove other trigonometric identities. 1 Pre-Calculus 12A Section 6.1, 6.3, 6.4 Example 1: Verify a Potential Identity Numerically Consider sec tan sin a. Determine the non-permissible values, in degrees, for the identity sec b. Numerically verify that 60 and tan c. Prove that sec is an identity. sin 4 tan sin are solutions to the equation sec tan . sin Solution: a. To determine the non-permissible values, assess each trigonometric function in the equation individually and examine expressions that may have non-permissible values. Non-permissible values of the variable are values for which denominators are zero or numerators are undefined. sec Since sec undefined. tan sin Left Hand Side Right Hand Side Consider sec tan 1 , cos 0 as division by 0 is cos tan is not defined at 90 180n . Thus 90 180n are non-permissible values for the variable. What values of would cause cos 0 ? The non-permissible values of would be ________________________. sin Since the denominator on the RHS is sin , you must also consider what values of would cause sin 0 . The non permissible values of are; ______________________________________. Combined, the non-permissible values for are: ______________________________________________ 2 Pre-Calculus 12A Section 6.1, 6.3, 6.4 b. Verify the equation sec 60 sec sec 60 1 cos 60 1 0.5 2 sec tan is valid for x = 60 and sin 4 tan sin tan sin tan 60 sin 60 3 3 2 2 LHS=RHS The equation is true for 60 . 4 sec 4 1 cos 4 1 1 2 2 sec sec tan sin tan sin 4 sin 4 1 1 2 2 tan LHS=RHS The equation is true for 4 . Verifying that an equation is valid for a particular value of the variable is not sufficient to conclude that the equation is an identity. c. Prove that sec tan is an identity. sin To prove that an identity is true for all permissible values, express both sides of the identity in equivalent forms. One or both sides of the identity must be algebraically manipulated into an equivalent form to match the other side. There is a major difference between solving a trigonometric equation and proving a trigonometric identity: Solving a trigonometric equation determines the value(s) that make that particular equation true. You perform operations across the == sign to solve for the variable. Proving a trigonometric identity shows that the expressions on each side of the =sign are equivalent. You work on each side of the identity independently, and you do not perform operations across the = sign. 3 Pre-Calculus 12A Section 6.1, 6.3, 6.4 sec tan sin LHS sec RHS tan sin Example 2: Use Identities to Simplify Expressions a. Determine the non-permissible values, in radians, of the variable in the expression b. Simplify the expression. tan x cos x sec x cot x Solution: a. The non-permissible values of tan x are___________________________________ The non-permissible values of sec x are _________________________________ The non-permissible values of cot x are __________________________________. Combined, the non-permissible values for tan x cos x are ____________________ sec x cot x b. To simplify the expression, use the reciprocal and quotient identities to rewrite the expression in terms of cosine and sine. Simplify tan x cos x sec x cot x 4 Pre-Calculus 12A Section 6.1, 6.3, 6.4 Example 3: Using the Reciprocal and Quotient Identities Prove each of the following identities for all permissible values of x. tan x cos x cos2 x sin x a. b. cot x sec x 1 sin x sin x cos x Solution: a. b. LHS tan x cos x sin x tan x cos x 1 sin x cot x sec x RHS 1 LHS cot x sec x cos2 x sin x sin x cos x RHS 2 cos x sin x sin x cos x The Pythagorean Identity Development of the Pythagorean Identity: In the diagram to the left note the Pythagorean Theorem: a2 b 2 c 2 . Replace these values with x, y and r. ____________________________________ In the unit circle r = 1, cos x and sin y , substitute these values for x, y and r. THE PYTHAGOREAN IDENTITY__________________________________________ cos2 sin2 1 cos2 sin2 5 Pre-Calculus 12A Section 6.1, 6.3, 6.4 cos2 sin2 1 cos2 sin2 1 Divide each term by cos2 Divide each term by sin2 Now isolate each term individually to find the other versions of the identity. Now isolate each term individually to find the other versions of the identity. THE PYTHAGOREAN IDENTITIES Example 4: Using the Pythagorean Identities Prove each of the following identities for all permissible values of x. a. cos x sin2 x cos x cos3 x b. sec x cos x tan x sin x c. tan x sin x sec x cos x sec x d. sin x cot x tan x Solution: a. b. cos x sin2 x cos x cos3 x LHS cos x sin2 x cos x RHS cos3 x sec x cos x tan x sin x LHS sec x RHS cos x tan x sin x 6 Pre-Calculus 12A Section 6.1, 6.3, 6.4 c. d. tan x sin x sec x cos x LHS tan x sin x RHS sec x cos x sin x LHS sin x sec x cot x tan x RHS sec x cot x tan x Tips for Proving Trigonometric Identities It is easier to simplify a more complicated expression than to make a simple expression more complicated, so start with the more complicated side of the identity. Use known identities to make substitutions. Rewrite the expressions using sine and cosine only. If a quadratic is present; consider the Pythagorean identities. Factor expressions to cancel out terms or create other identities. Expand expressions to create identities. If you have fractions, think about making common denominators. Multiply the numerator and denominator by the conjugate of an expression. Always keep the “target expression” in mind. Continually refer to it. 7 Pre-Calculus 12A Section 6.1, 6.3, 6.4 Example 5: Solving Equations Using the Quotient, Reciprocal, and Pythagorean Identities Solve for in general form and then in the specified domain. Use exact solutions when possible. Check for non-permissible values. a. cos2 cot sin , 0 2 b. 2sin 7 3csc , 360 180 c. sin2 2cos 2, 2 2 Solution: a. cos2 cot sin , 0 2 b. 2sin 7 3csc , 360 180 c. sin2 2cos 2, 2 2 8