The Unit Circle |

**THE UNIT CIRCLE :**

The unit circle is a circle, center (0, 0) with a radius of 1 unit.

The** **** x** and

since it is a ray of the positive

**Note:**

- sin

on the unit circle and

- cos

on the unit circle and

- tan

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, lim_{x }_{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 .

.

**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

. |

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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 |

.

2/ State 2 coterminal angles for:

- a) P(A) = (0, – 1)

- b) .

.

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 .

.

- 3/ Evaluate:

a) = 2(1) + 3(–½) – 5(–1) =

- b)

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