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Stitz-Zeager_College_Algebra_e-book

# Our next batch of identities makes heavy use of the

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Unformatted text preview: √ √ 1 , − 23 2 (0, −1) 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. 5 1. Proceeding as in Example 10.2.2, we substitute cos(α) = 13 into cos2 (α) + sin2 (α) = 1 and 12 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 13 5 x-axis to the ray which contains the point (cos(α), sin(α)) = 13 , 12 . 13 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 5 12 The Reference Angle Theorem, cos(θ) = ± cos(α) = ± 13 and sin...
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