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Stitz-Zeager_College_Algebra_e-book

# Since we are given x cost we know sect 1 1 x the

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Unformatted text preview: sin(β ) 10.4 Trigonometric Identities 659 Example 10.4.2. 1. Find the exact value of sin 19π 12 2. If α is a Quadrant II angle with sin(α) = ﬁnd sin(α − β ). 5 13 , and β is a Quadrant III angle with tan(β ) = 2, 3. Derive a formula for tan(α + β ) in terms of tan(α) and tan(β ). Solution. 1. As in Example 10.4.1, we need to write the angle 19π as a sum or diﬀerence of common angles. 12 The denominator of 12 suggests a combination of angles with denominators 3 and 4. One π such combination is 19π = 43 + π . Applying Theorem 10.15, we get 12 4 sin 19π 12 4π π + 3 4 4π π 4π π = sin cos + cos sin 3 4 3 4 √ √ √ 3 2 1 2 = − +− 2 2 2 2 √ √ − 6− 2 = 4 = sin 2. In order to ﬁnd sin(α − β ) using Theorem 10.15, we need to ﬁnd cos(α) and both cos(β ) and sin(β ). To ﬁnd cos(α), we use the Pythagorean Identity cos2 (α) + sin2 (α) = 1. Since 5 52 sin(α) = 13 , we have cos2 (α) + 13 = 1, or cos(α) = ± 12 . Since α is a Quadrant II angle, 13 cos(α)...
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