Quiz 10 - Math 1022 Name:
Directions: Complete these problems on this paper or separate paper and show all work. Afterwards,
take a picture of your solutions and convert the pictures into a single .pdf file. You can use an app such
as Adobescan, camscanner, or iphone notes to do this easily. Go to Canvas →Assignments and find this
assignment and upload your .pdf file there and submit it.
1. Compute arctan(tan(5π/4))
We have 5π
4is outside the range of arctan, which is (−π/2, π/2). However, we know
tan 5π
4= 1 = tan π
4
Since π
4is in the range of arctan, we have that
arctan(tan(5π/4)) = π/4.
2. Compute sin(arctan(2/3))
Set θ= arctan(2/3). Then θlies in a right triangle with opposite 2 and adjacent 3, thus hypotenuse
equal to √4 + 9 = √13. So
sin(arctan(2/3)) = sin(θ) = 2
√13.
3. Simplify tan(arcsin(x/4)).
Set θ= arcsin(x/4). Then θis the interior angle of a right triangle with opposite x, hypotenuse 4,
and adjacent √16 −x2. Thus
tan(arcsin(x/4)) = tan(θ) = x
√16 −x2.
4. Simplify cos(2 arccos(x))
Set θ= arccos(x). Then θis the interior angle of a right triangle with adjacent x, hypotenuse 1, and
opposite √1−x2. We have
cos(2 arccos(x)) = cos(2θ) = cos2(θ)−sin2(θ) = x2
1−1−x2
1= 2x2−1.
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