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Remove square root without using calculator

Question:

Without using a calculator, find square root of 1 - sin 20 degrees.

Solution:

Using the formula: sin2x = 2 sin x cos x, so, sin 20 degrees = 2 sin 10 degrees cos 10 degrees.

square root of 1 - sin 20 degrees = square root of 1 - 2 sin 10 degrees cos 10 degrees

using the formula: sin2x + cos2x = 1, so, sin210 degrees + cos210 degrees = 1

square root of 1 - sin 20 degrees
= square root of 1 - 2 sin 10 degrees cos 10 degrees
= square root of sin210 degrees + cos210 degrees - 2 sin 10 degrees cos 10 degrees
= square root of sin210 degrees - 2 sin 10 degrees cos 10 degrees + cos210 degrees

using the formula: (a - b)2 = a2 - 2ab + b2

then
square root of sin210 degrees - 2 sin 10 degrees cos 10 degrees + cos210 degrees
= square root of (sin 10 degrees - cos 10 degrees)2

Note: the number inside square root needed to be positive or zero

square root of (sin 10 degrees - cos 10 degrees)2
= |sin 10 degrees - cos 10 degrees|

Because sin 10 degrees is less than cos 10 degrees, so,

|sin 10 degrees - cos 10 degrees|
= cos 10 degrees - sin 10 degrees

Therefore, the square root of 1 - sin 20 degrees = cos 10 degrees - sin 10 degrees. Watch the video for more details.