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How to Memorize the Unit Circle?

TopA Circle has radius as one of its characteristics. Although, radius can be of any measure, but when it is equals to 1, circle is said to be a Unit Circle. How to memorize the unit circle as fast as possible? For this let us discuss few important points regarding unit circle which are as follows:
Unit circle can be best remembered in terms of lengths and angles from different perspectives. Suppose we have a unit circle with center at Intersection Point of 'x' and y- axes. Its graph is given as:

Equation of unit circle is given as X2 + Y2 = 1. For different angles we have different behaviors of equation according to following diagram: The equation of the unit circle is given as X2 + Y2 = 1. For different angles we have different behaviors of the equation according to the following diagram:


1. At 45 degrees, Y = X
X2 + X2 = 1,
2X2 = 1,
X2 = 1 / 2,
X = Y = √ (1 / 2).
2. At 60 degrees all sides of triangle are equal: X = 1 / 2 And "Y" can be calculated as:
(1/2)2 + Y2 = 1,
1/4 + Y2 = 1,
Y2 = 1 - 1/4 = ¾,
Y = √3/4.
3. At 30 degrees X= √3/4 and Y = ½.
According to pythagoras theorem we can write:
X2 + Y2 = 12,
or X2 + Y2 = 1,
Polar form coordinates of unit circle can be written as: X = cos A and Y = sin A. Substituting their values 'I' equation of the unit circle we get:
(cos(θ))2 + (sin(θ))2 = 1 [It’s a very useful trigonometric identity].
For basic Trigonometric Functions cosine, sine and Tangent we define the value of Functions at some known angle as:

Angle

Sine

Cosine

Tangent

300

½

√3 / 2

1/√3

450

1/√2

1/√2

1

600

√3 / 2

½

√3