Unit Circle

SAT Questions that focus on Unit Circle require knowledge of the following topics.

Units used to represent angles

Degrees and radians are the most common units used to represent angles.
In order to convert one unit to the other, remember that one turn measures:

• 360º (Degrees).
• $2\pi$ (in Radians) (angles expressed in radians are dimensionless)
• 400gr (Grads)

Radians

1 radian is defined in a circle as the angle whose arc is equal to one circle's radius (following figure).

One turn is 2π radians. The number π is a constant, that equals approximately 3.14159265359 (in tests it is usually enough to use only three digits: 3.14).

Unit Circle

The unit circle is a circle with radius one and centered at the origin (0, 0) of the Cartesian coordinate system. In this system, the x-axis is the cosine, and the y-axis is the sine (figure below):

Arcs measured counter clockwise in the Unit Circle are positive; arcs measured clockwise are negative.

The starting point in the Unit Circle is the cosine-axis (x-axis). This is the point where lies angle 0º.

Each turn in the Unit Circle has 360º. Thus, all angles x+n360º fall on the same point of the Unit Circle (x is any angle between 0 and 360º, and "n" is an integer). For example, 30º and 390º fall on the same point of the Unit Circle.

Since all angles x+n360º fall on the same point of the Unit Circle, they all have the same values for sine, cosine, tangent, etc.

Note that the Unit Circle has four quadrants, I, II, III e IV (in red roman letters in the figure above).

The sign of a trigonometric function depends on the quadrant the angle falls in:
In quadrant I, sine and cosine functions are positive.
In quadrant II, sine is positive and cosine is negative.
In quadrant III, sine and cosine functions are negative.
In quadrant IV, sine is negative and cosine is positive.

Solved SAT Practice Tests

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SAT Practice Tests - Unit Circle

Additional Practice Tests - Unit Circle