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**Unformatted text preview: **cotangent. We 10.7 Trigonometric Equations and Inequalities 731 x)
x)
choose3 to use the quotient identity cot(3x) = cos(3x) . Graphing y = cos(3x) and y = 0 (the
sin(3
sin(3
x-axis), we see that the x-coordinates of the intersection points approximately match our
solutions. 4. To solve sec2 (x) = 4, we ﬁrst extract square roots to get sec(x) = ±2. Converting to cosines,
1
π
we have cos(x) = ± 2 . For cos(x) = 1 , we get x = π + 2πk or x = 53 + 2πk for integers k .
2
3
2π
4π
1
For cos(x) = − 2 , we get x = 3 + 2πk or x = 3 + 2πk for integers k . Taking a step back,4
π
π
we realize that these solutions can be combined because π and 43 are π units apart as are 23
3
5π
π
2π
and 3 . Hence, we may rewrite our solutions as x = 3 + πk and x = 3 + πk for integers k .
Now, depending on the integer k , sec π + πk doesn’t always equal sec π . However, it is
3
3
true that for all integers k , sec π + πk = ± sec π = ±2. (Can you show this?) As a result,
3
3
π
sec2 π + πk = (±...

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