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Unformatted text preview: 4.08 SECTIONS 4.24.4: TRIG FUNCTIONS (VALUES AND IDENTITIES) We will consider two general approaches: the Right Triangle approach, and the Unit Circle approach. PART A: THE RIGHT TRIANGLE APPROACH The Setup The acute angles of a right triangle are complementary. Consider such an angle, θ . Relative to θ , we may label the sides as follows: The hypotenuse always faces the right angle, and it is always the longest side. The other two sides are the legs. The opposite side (relative to θ ) faces the θ angle. The other leg is the adjacent side (relative to θ ). We sometimes use the terms “hypotenuse,” “leg,” and “side” when we are actually referring to a length. 4.09 Defining the Six Basic Trig Functions (where θ is acute) The Ancient Curse (or “How to Define Trig Functions”) SOHCAHTOA Sine θ = sin θ = Opp. Hyp. Cosine θ = cos θ = Adj. Hyp. Tangent θ = tan θ = Opp. Adj. Reciprocal Identities (or “How to Define More Trig Functions”) Cosecant θ = csc θ = 1 sin θ = Hyp. Opp. ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ Secant θ = sec θ = 1 cos θ = Hyp. Adj. ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ Cotangent θ = cot θ = 1 tan θ = Adj. Opp. ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ Warning: Remember that the reciprocal of sin θ is csc θ , not sec θ . Note: We typically treat “0” and “undefined” as reciprocals when we are dealing with trig functions. Your algebra teacher will not want to hear this, though! 4.10 Quotient Identities We may also define tan θ and cot θ as follows: Quotient Identities tan θ = sin θ cos θ and cot θ = cos θ sin θ Why is this consistent with SOHCAHTOA? sin θ cos θ = Opp. Hyp. Adj. Hyp. = Opp. Adj. = tan θ cot θ is the reciprocal of tan θ . We will discuss the Cofunction Identities soon …. 4.11 The Pythagorean Theorem Given two sides, you can find the length of the third by using the Pythagorean Theorem (see p.349 for one of many proofs): Opp. ( ) 2 + Adj. ( ) 2 = Hyp. ( ) 2 Pythagorean Triples Pythagorean triples are a set of three integers that can represent the side lengths of a right triangle. The most famous Pythagorean triples are: 345 51213 81517 Some less famous ones are: 72425 94041 Warning: Remember that the hypotenuse must be the longest side. If the two legs of a right triangle have lengths 3 and 5, the hypotenuse is not 4....
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 Fall '11
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 Calculus, Angles, Pythagorean Theorem, Unit Circle, Cos, Hypotenuse, triangle, tan θ

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