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Q.

Find the reference angle for 108 $°$ .

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a

ρ=50

°

b

ρ=60

°

c

ρ=72

°

d

ρ=40

°

answer is C.

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Detailed Solution

We try to break the angles inside 360

°

into the four normal quadrants. There are 4 quadrants - first, second, third, fourth. Now for any kind of given angle αα, the reference angle ρ will be the acute angle that the terminal side of the angle α makes with the x-axis.
These angles generally have non-quadrantal angles in normal positions. We always need to find the difference between the angle and the positive or negative x-axis.
Now we express the quadrant with different results.
For given angle α in the first quadrant (0

°

<α<90

°

), the reference angle will be ρ=α.
For given angle α in the second quadrant (90

°

<α<180

°

), the reference angle will be ρ=180

°

−α=π−α.
For given angle αα in the third quadrant (180

°

<α<270

°

), the reference angle will be ρ=α−180

°

=α−π.
For given angle αα in the first quadrant (270

°

<α<360

°

), the reference angle will be ρ=360

°

−α=2π−α
ρ=360

°

−α=2π−α.
The representation will be of the form
Question Image

Question Image

For the given problem we need to find a reference angle for 108∘.
The angle 108

°

is in the second quadrant as (90

°

<108

°

<180

°

).
So, using the formulas the reference angle will be π−108

°

=180

°

−108

°

=72

°

.
So, the correct answer is “Option 3”.

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