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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 √ =± and with y = x we have 2
1 P (x, y ) P (x, y ) θ = 45◦ 45◦
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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