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Pre-Calculus 12A Section 6.1 Reciprocal, Quotient, and Pythagorean Identities It is critical to have a clear understanding of the difference between equations and identities. Equations can be solved for certain value(s) of the variable. For example, the equation 3x = 12 is only true when x = 4. Identity: an identity is an equation that is true for all values of the variables. For example: x y 2 x 2 2xy y 2 This equation is true for all possible values of x and y, so it is called an identity. Trigonometric identity: a trigonometric equation that is true for all permissible values of the variable in the expressions on both sides of the equation. Non-permissible values of trigonometric identities occur when there is a denominator in the identity. The nonpermissible values of any trigonometric identity can be determined by finding all values of that would cause a denominator of zero. 1 , is not defined at sin 0 which occurs at 0 and 180 sin and every subsequent rotation of 180 . Therefore the non-permissible values for this identity in degrees would be 0 180n, n I . In radians 0 n,n I . For example, the trigonometric identity, csc 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 Example 1: Verify a Potential Identity Numerically a. Determine the non-permissible values, in degrees, for the equation sec b. Numerically verify that 60 and c. Verify the identity algebraically. 4 tan sin are solutions of the equation. 1 Pre-Calculus 12A Section 6.1 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. 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 . So the nonpermissible vales of are __________________ 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; ______________________________________. The three sets of non permissible values can be expressed as ______________________________________________ b. Verify numerically. tan sec 60 sin sec tan sin sec 60 tan 60 1 sin 60 cos 60 3 1 0.5 3 2 2 2 LHS=RHS 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 2 Pre-Calculus 12A Section 6.1 c. Verify algebraically. sec 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 identity for sec x and the quotient identities for tan x and cot x to write the trigonometric expressions in terms of cosine and sine. Simplify tan x cos x sec x cot x 3 Pre-Calculus 12A Section 6.1 Example 3: Using the Reciprocal and Quotient Identities Prove each of the following identities: tan x cos x a. 1 sin x b. cot x sec x cos2 x sin x sin x cos x Solution: tan x cos x sin x 1 cot x sec x cos2 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 4 Pre-Calculus 12A Section 6.1 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: 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: cos x sin2 x cos x cos3 x sec x cos x tan x sin x 5 Pre-Calculus 12A tan x sin x Section 6.1 sec x cos x sin x sec x cot x tan x Strategies That May Assist in Proving Trigonometric Identities 1. Start by working on whatever side looks more complicated. Then work toward the less complicated expression. 2. Rewrite everything in terms of sine and cosine. 3. If you have fractions, think about making common denominators. 4. Factor expressions to cancel out terms or create other identities. 5. Expand expressions to create identities. 6. When there is a square in the term, look for a way to apply the Pythagorean Identity. 7. Always keep the “target expression” in mind. Continually refer to it. 6