6.1: Reciprocal, Quotient & Pythagorean Identities

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1 Math Pre-Calculus 6.: Reciprocal, Quotient & Pythagorean Identities A trigonometric identity is an equation that is valid for all values of the variable(s) for which the equation is defined. In this chapter we will verify several trigonometric identities using other known identities (equations). Some identities you already know: Reciprocal Identities csc x sec x cot x tan x Using the diagram of the unit circle provided and the definitions of trigonometric functions, determine the following basic trigonometric identities. Pythagorean Identities Quotient Identities Verifying Identities We can verify identities by two methods: graphically and numerically. The only way that we can prove that an equation is actually an identity (true for all values) is algebraically.

2 Math Pre-Calculus Ex. : a) Determine the non-permissible values, in degrees, for the equation tan sec. sin b) Numerically verify that 60 and are solutions of the equation. 4 c) Use technology to graphically decide whether the equation could be an identity. Ex. : a) Determine the non-permissible values, in radians, of the variable in the expression cot x csc xcos x b) Simplify the expression as much as possible.

3 Math Pre-Calculus Ex. 3: a) Verify that the equation cot x csc x is true when x. 6 b) Show that the Pythagorean identity is equivalent to cot x csc x 6.: Sum, Difference & Double-Angle Identities The Sum Identities are: sin( A ) sin Acos cos Asin cos( A ) cos Acos sin Asin tan A tan tan( A) tan Atan The Difference Identities are: sin( A ) sin Acos cos Asin cos( A ) cos Acos sin Asin tan A tan tan( A) tan Atan Ex. : Write each expression as a single trigonometric function. a) sin 48 cos7 cos 48 sin7 b) cos88 cos35 sin88 sin35 3

4 Math Pre-Calculus Ex. : Consider each of the sum identities for =A to derive the double-angle identities The Double-Angle Identities are: sin A sin Acos A tan A tan A tan A cos cos sin A A A cos Acos A cos A sin A Ex. 3: Consider the expression cos x. a) What are the permissible values for the expression? b) Simplify the expression to one of the three primary trigonometric functions. 4

5 Math Pre-Calculus Ex. 4: Determine the exact value for the expression sin. 6.3: Proving Identities Investigate Work on the left side of this equation until it equals the right side. Hints are given for each step. Step Hint L.S. (Left Side) R.S. (Right Side). use tan x. combine L.S. terms with a common denominator tan x sec x 3. use sin x 4. use sec x Notes: Restrictions on the variable(s) in identities are understood to apply. They are not usually specified unless requested. We will use a two-column proof format to prove identities. The idea is to simplify one or both sides until they are equal. We may use all of the identities presented so far. Keep your formula sheet handy! 5

6 Math Pre-Calculus Ex. : Prove each of the following identities. State your reasoning along the way. tan x cot x 6

7 Math Pre-Calculus Some hints for proving identities: It is often helpful to rewrite everything in terms of sine and cosine (use the quotient identities). We will often need to write everything over a common denominator. Try to match L.S. and R.S. Multiply the numerator and denominator by the conjugate of an expression. Where possible, factor a GCF or a trinomial. Practice as many different proofs as possible. The more you have seen, the easier they will be! If you are stuck sometimes the best thing to do is erase and start over Ex. : Prove the following identity for all permissible values of x. cot x csc x 6.4: Solving Trigonometric Equations Using Identities We have solved trigonometric equations before (Chapter 4). Sometimes, for more complex equations we will need to use identities to simplify an equation to only one trigonometric function. We can verify answers on our calculator, or by substitution. We are again solving over a specified domain, or asked for a general solution. Always check for npv s (or check by substitution). 7

8 Math Pre-Calculus Ex. : Solve each equation algebraically over the domain 0x. a) 0 b) 3 Ex. : Solve the equation answer. cos x cot xsin x algebraically in the domain 0 x 360. Verify your Ex. 3: Solve the equation algebraically. Give the general solution expressed in radians. 8

9 Math Pre-Calculus Ex. 4: Algebraically solve 7 3csc x. Give general solutions expressed in radians. 9