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Stitz-Zeager_College_Algebra_e-book

# Y y 1 4 2 3 4 3 4 x 2 1 4 1 1 reect across

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Unformatted text preview: = − 12 . We now set about ﬁnding cos(β ) and sin(β ). We have several ways to proceed, 13 but the Pythagorean Identity 1 + tan2 (β ) = sec2 (β ) is a quick way to get sec(β ), and hence, √ cos(β ). With tan(β ) = 2, we get 1 + 22 = sec2 (β ) so that sec(β ) = ± √5. Since β is a √ 1 1 Quadrant III angle, we choose sec(β ) = − 5 so cos(β ) = sec(β ) = −√5 = − 55 . We now need to determine sin(β ). We could use The Pythagorean Identity cos2 (β ) + sin2 (β ) = 1, but we sin( opt instead to use a quotient identity. From tan(β ) = cos(β ) , we have sin(β ) = tan(β ) cos(β ) β) √ so we get sin(β ) = (2) − 5 5 √ = − 2 5 5 . We now have all the pieces needed to ﬁnd sin(α − β ): sin(α − β ) = sin(α) cos(β ) − cos(α) sin(β ) √ √ 5 5 12 25 = − −− − 13 5 13 5 √ 29 5 =− 65 660 Foundations of Trigonometry 3. We can start expanding tan(α + β ) using a quotient identity and our sum formulas tan(α + β ) = = sin(α + β ) cos(...
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