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. To solve sec(θ) = 2, we convert to cosines and get cos(θ) = 2 or cos(θ) = 1 . This is the exact
same equation we solved in Example 10.2.5, number 1, so we know the answer is: θ = π + 2πk
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
know the solutions to tan(θ) = 3 must, therefore, have a reference angle of π . Our next
task is to determine in which quadrants the solutions to this equation lie. Since tangent is
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
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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