Stitz-Zeager_College_Algebra_e-book

The angle of elevation from the end of the shadow to

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Unformatted text preview: . When we sketch θ = 45◦ in standard position, we see that its terminal does not lie along any of the coordinate axes which makes our job of finding the cosine and sine values a bit more difficult. Let P (x, y ) denote the point on the terminal side of θ which lies on the Unit Circle. By definition, x = cos (45◦ ) and y = sin (45◦ ). If we drop a perpendicular line segment from P to the x-axis, we obtain a 45◦ − 45◦ − 90◦ right triangle whose legs have lengths x and y units. From Geometry, we get y = x.2 Since P (x, y ) lies on the Unit Circle, we have x2 + y 2 = 1. Substituting y = x into this equation yields 2x2 = 1, or x = ± Since P (x, y ) lies in the first quadrant, x > 0, so x = cos (45◦ ) = y= sin (45◦ ) √ = √ 2 2 1 2 √ =± and with y = x we have 2 2. y 1 P (x, y ) P (x, y ) θ = 45◦ 45◦ x 1 θ = 45◦ x 2 Can you show this? 2 2. y 614 Foundations of Trigonometry 4. As before, the terminal side of θ = π does not lie on any of the coordinate axes, so we proce...
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