As the reader can verify a phase shift of to the left

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Unformatted text preview: ce of identities in trigonometry. Our next task is to use use the Reciprocal and Quotient Identities found in Theorem 10.6 coupled with the Pythagorean Identity found in Theorem 10.1 to derive new Pythagorean-like identities for the remaining four circular functions. Assuming cos(θ) = 0, we may start with cos2 (θ) + sin2 (θ) = 1 and divide both sides sin2 by cos2 (θ) to obtain 1 + cos2(θ) = cos1(θ) . Using properties of exponents along with the Reciprocal 2 (θ) and Quotient Identities, reduces this to 1 + tan2 (θ) = sec2 (θ). If sin(θ) = 0, we can divide both sides of the identity cos2 (θ) + sin2 (θ) = 1 by sin2 (θ), apply Theorem 10.6 once again, and obtain cot2 (θ) + 1 = csc2 (θ). These three Pythagorean Identities are worth memorizing, and they are summarized in the following theorem. 640 Foundations of Trigonometry Theorem 10.8. The Pythagorean Identities: • cos2 (θ) + sin2 (θ) = 1. • 1 + tan2 (θ) = sec2 (θ), provided cos(θ) = 0. • cot2 (θ) + 1 = csc2 (θ), provided sin(θ) = 0. Trigonometric identities play an important role in not just Trigonometry, but in Calculus as well. We’ll use them in this book to ...
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