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Unit 4 Section 6

# Unit 4 Section 6 - 4.6 FUNDAMENTAL IDENTITIES Recall the...

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4.6 FUNDAMENTAL IDENTITIES Recall the wrapping function such that: where t is the arc length of an arc in standard position and is the terminal point. Then, we define the following circular functions as: Recall that we define an identity as an equation that is always true for all possible values of the variable(s) involve. From our discussions of circular functions, we can obtain several identities from definitions and some algebraic manipulations. Eight Fundamental Identities Quotient Relationships: and Reciprocal Relationships: Pythagorean Relationships: Since the points on the unit circle satisfies the equation we have the Pythagorean Identity . Dividing both sides of the equation by , we have Dividing both sides by , we get . As a summary, the eight fundamental identities are as follows: Example 4.6.1 Find the other circular function values of . 1.

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Solution: We can immediately evaluate . By the Pythagorean identity, we have Since , then . Therefore, . . 2. Solution: Since and , then . NO we can’t! Why? . . Proving/Verifying an Identity Proving or verifying an identity is a way to show that two quantities are always equal for all possible values of the variable involved. The following statements give us the general guidelines in proving an identity that involves circular functions. 1. Know the eight basic relationships and recognize alternative forms of each. 2. Recall the procedures for adding and subtracting, reducing, and transforming rational expressions into equivalent forms. 3. Remember factoring and special product techniques.
4. Use only substitution and simplification procedures that allow you to work on exactly one side of an equation. 5.

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Unit 4 Section 6 - 4.6 FUNDAMENTAL IDENTITIES Recall the...

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