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Important Points on the Unit Circle
6 For once, we have something convenient about using radian measure in contrast to the abstract theoretical
nonsense about using them as a ‘natural’ way to match oriented angles with real numbers! 620 Foundations of Trigonometry The next example summarizes all of the important ideas discussed thus far in the section.
Example 10.2.4. Suppose α is an acute angle with cos(α) = 5
13 . 1. Find sin(α) and use this to plot α in standard position.
2. Find the sine and cosine of the following angles:
(a) θ = π + α (b) θ = 2π − α (c) θ = 3π − α (d) θ = π
2 +α Solution.
1. Proceeding as in Example 10.2.2, we substitute cos(α) = 13 into cos2 (α) + sin2 (α) = 1 and
ﬁnd sin(α) = ± 13 . Since α is an acute (and therefore Quadrant I) angle, sin(α) is positive.
Hence, sin(α) = 12 . To plot α in standard position, we begin our rotation on the positive
x-axis to the ray which contains the point (cos(α), sin(α)) = 13 , 12 .
y 1 5 12
13 , 13 α
1 x Sketching α
2. (a) To ﬁnd the cosine and sine of θ = π + α, we ﬁrst plot θ in standard position. We can
imagine the sum of the angles π + α as a sequence of two rotations: a rotation of π radians
followed by a rotation of α radians.7 We see that α is the reference angle for θ, so by
The Reference Angle Theorem, cos(θ) = ± cos(α) = ± 13 and sin...
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