The Unit Circle

THE UNIT CIRCLE :

The unit circle is a circle, center (0, 0) with a radius of 1 unit.
The x and y coordinates of the points on this circle are the cosine and sine of the angle formed between the radius of the circle and the positive x-axis,
since it is a ray of the positive x-axis that we rotate to create our angles.

Note:

sin u > 0 in the 1st & 2nd quadrants since sin u is the y value at any point
on the unit circle and y > 0 in those quadrants.
cos u > 0 in the 1st & 4th quadrants since cos u is the x value at any point
on the unit circle and x > 0 in the 1st and 4th quadrants.
tan u > 0 when x and y have the same sign, ie: in the 1st and 3rd quadrants.

 cos 0 = 1 sin 0 = 0 tan 0 = 0 cos = 0 sin = 1 tan cos = – 1 sin = 0 tan = 0 cos = 0 sin = – 1 tan

the values for are the same as those for 0.

(Calculus Students: When doing limits, just follow these values. That is, limx d 0 cos x = 1.)

Thus, the unit circle maps each angle's cosine and sine values onto a point.
We can now write the coordinates of a point (
x, y) as (cos A , sin A ).
As we see in our unit circle, the cosine function has a domain of R, and a range of [ –1, 1 ].

The values for 0, , , and , are read from the points (1, 0), (0, 1), ( – 1, 0) and (0, – 1) respectively as listed in the table.

The values for cosine and sine of the other angles shown are read from
the 30°, 60°, 90° and the 45°, 45°, 90° triangle.

Note: we generally express the sine and the cosine of 45° as rather than .

In the diagram above, we see the central angles, however, we do not see the coordinates of the points. Those would be (cos A , sin A ).

First Quadrant Points on the Unit Circle:

From the symmetry of the unit circle, and our knowledge of the signs
for the x and y values, we can find the remaining coordinates.

use the triangles to find the coordinates of the 12 points on the circle.

Hints and Notes:

Note1: A full rotation of is a full circle.

So an angle of is in the 1st quadrant since it's radians greater than a full circle.

Note2: To locate an angle of , just count your way to the proper quadrant. There are 3 angles of per quadrant so takes us to the 2nd quad. leaving one angle of .

Note3: We must adjust the sign of the trig function according to the angle's quadrant, so we first decide the value of the trig function and then determine the sign. (positive or negative)

Example 1:

The coordinates of P(A), a point on the unit circle are .

Find 2 coterminal angles for A.

Obviously we've got an angle of 45° or radians. Since the y-value is < 0,
but x > 0 we know it's in the 4th quad.
It takes 7 angles of 45° or radians to reach the 4th quad,
so the angle is either or . We could also have used or .

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Example 2:

Find the 6 trig function values for angle A if

Since we have a multiple of , we know the 6 values, from the 30, 60, 90 triangle. We just need to adjust the sign to fit the quadrant. more than a complete rotation clockwise. Which puts us in the 3rd quadrant where only tan A is positive

 .

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Practice

1/ Complete the following table without a calculator: do not rationalize!!

(you're allowed to copy and paste the table so you can print it out for efficiency -- just the table!!)

Hint: -- divide by zero, get infinity.

 A sin A cos A tan A csc A sec A cot A 0

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2/ State 2 coterminal angles for:

a) P(A) = (0, – 1)
b) .

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3/ Evaluate:

a)
b)

Solutions

1/

 A sin A cos A tan A csc A sec A cot A –1/2 –2 –1 0 –1 0 –1/2 –2 –1 –1 0 –1 0 –1 0 0 –1 0 1

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2/

a) or .
b) or .

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3/ Evaluate:
a) = 2(1) + 3(–½) – 5(–1) = 5.5
b)

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(all content of the MathRoom Lessons © Tammy the Tutor; 2004 - ).