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Unformatted text preview: ine and cosine are negative (Can you see why?) so we
conclude sin(θ) = − 2 5 5 . 3. When we substitute sin(θ) = 1 into cos2 (θ) + sin2 (θ) = 1, we ﬁnd cos(θ) = 0.
Another tool which helps immensely in determining cosines and sines of angles is the symmetry
inherent in the Unit Circle. Suppose, for instance, we wish to know the cosine and sine of θ = 56 .
We plot θ in standard position below and, as usual, let P (x, y ) denote the point on the terminal
side of θ which lies on the Unit Circle. Note that the terminal side of θ lies π radians short of one
half revolution. In Example 10.2.1, we determined that cos
6 √ = 3
2 This is unfortunate from a ‘function notation’ perspective. See Section 10.6.
See Sections 1.1 and 7.2 for details. and sin π
6 = 1 . This means
2 616 Foundations of Trigonometry
√ that the point on the terminal side of the angle π , when plotted in standard position, is 23 , 1 .
From the ﬁgure below, it is clear that the point P (x, y ) we seek can be obtained by reﬂecting that
point about the y -axis. Hence, cos 56 = − 23 and sin 56 = 1 .
y y 1 1 θ= P (x, y ) √
2 √ 5π
6 P− π
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