Stitz-Zeager_College_Algebra_e-book

676 foundations of trigonometry a 2x a x 0 2x

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Unformatted text preview: xample 10.3.2. Find all angles which satisfy the given equation. 1. sec(θ) = 2 2. tan(θ) = √ 3 3. cot(θ) = −1. Solution. 1 1. To solve sec(θ) = 2, we convert to cosines and get cos(θ) = 2 or cos(θ) = 1 . This is the exact 2 same equation we solved in Example 10.2.5, number 1, so we know the answer is: θ = π + 2πk 3 π or θ = 53 + 2πk for integers k . √ 2. From the table of common values, we see tan π = 3. According to Theorem 10.7, we 3 √ know the solutions to tan(θ) = 3 must, therefore, have a reference angle of π . Our next 3 task is to determine in which quadrants the solutions to this equation lie. Since tangent is y defined as the ratio x , of points (x, y ), x = 0, on the Unit Circle, tangent is positive when x and y have the same sign (i.e., when they are both positive or both negative.) This happens in Quadrants I and III. In Quadrant I, we get the solutions: θ = π + 2πk for integers k , and 3 π for Quadrant III, we get θ = 43 + 2πk for integers k . While these descriptions of the solutions are correct, they can be combined into one list as θ = π + πk for integers k...
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