1.6 Trigonometric 1.6 Trigonometric Functions: The Unit circleFunctions: The Unit circle

The Unit Circle

A circle with radius of 1

Equation x2 + y2 = 1

sin,cos

0,1

1,0

0,1

1,0

Do you remember 30º, 60º, 90º triangles?

Do you remember 45º, 45º, 90º triangles?

Do you remember 45º, 45º, 90º triangles?

When the hypotenuse is 1

The legs are 2

2

1

2

2

2

2

Let's pick a point on the circle. We'll choose a point where the x is 1/2. If the x is 1/2, what is the y value?

(1,0)

(0,1)

(0,-1)

(-1,0)

x = 1/2

You can see there are two y values. They can be found by putting 1/2 into the equation for x and solving for y.

122 yx

12

1 22

y

4

32 y

2

3y

2

3,2

1

2

3,2

1

We'll look at a larger version of this and make a right triangle.

(1,0)

(0,1)

(0,-1)

(-1,0)

We know all of the sides of this triangle. The bottom leg is just the x value of the point, the other leg is just the y value and the hypotenuse is always 1 because it is a radius of the circle.

2

1

2

31

sin

cos2

1

121

Notice the sine is just the y value of the unit circle point and the cosine is just the x value.

tan 3

2123

2

3,2

12

3

123

(1,0)

(0,1)

(0,-1)

(-1,0)

2

3,2

1sin

cos

We divide the unit circle into various pieces and learn the point values so we can then from memory find trig functions.

tan

2

3,2

1

2

2

2

2

So if I want a trig function for whose terminal side contains a point on the unit circle, the y value is the sine, the x value is the cosine and y/x is the tangent.

2

2,

2

2

1

2222

Here is the unit circle divided into 8 pieces. Can you figure out how many degrees are in each division?

45°

We can label this all the way around with how many degrees an angle would be and the point on the unit circle that corresponds with the terminal side of the angle. We could then find any of the trig functions.

45°

2

2,

2

290°

1,0

0°

135°

2

2,

2

2

180° 0,1

225°

270°315°

2

2,

2

2

2

2,

2

2

1,0

225sin2

2

0,1

These are easy to

memorize since they all

have the same value with

different signs depending on the quadrant.

Can you figure out what these angles would be in radians?

The circle is 2 all the way around so half way is . The upper half is divided into 4 pieces so each piece is /4.

45°

2

2,

2

290°

1,0

0°

135°

2

2,

2

2

180° 0,1

225°

270°315°

2

2,

2

2

2

2,

2

2

1,0

4

7sin

2

2

0,14

2

4

3

4

5

2

34

7

Here is the unit circle divided into 12 pieces. Can you figure out how many degrees are in each division?

30°

We can again label the points on the circle and the sine is the y value, the cosine is the x value and the tangent is y over x.

30°

2

1,

2

3

90°

1,0

0°

120°

180° 0,1210°

270°

330°

1,0

330cos2

3

0,1

You'll need to memorize

these too but you can see the pattern.60°

150°

240°300°

2

3,2

1

2

3,2

1

2

3,2

1

2

1,

2

3

2

1,

2

3

2

1,

2

3

2

3,2

1240sin2

3

Can you figure out what the angles would be in radians?

30°

It is still halfway around the circle and the upper half is divided into 6 pieces so each piece is /6.

30°

2

1,

2

3

90°

1,0

0°

120°

180° 0,1210°

270°

330°

1,0

0,1

60°150°

240°300°

2

3,2

1

2

3,2

1

2

3,2

1

2

1,

2

3

2

1,

2

3

2

1,

2

3

2

3,2

1

We'll see them all put together on

the unit circle on the next screen.

6

You should memorize this. This is a great

reference because you can figure out

the trig functions of all these angles

quickly.

2

3,2

1

The Unit Circle with Radian Measures

2

The Six Trig functions

adjacent

opposite

b

aTan

hypotenuse

opposite

c

aSin

hypotenuse

adjacent

c

bCos

Cos

SinTan

Reciprocal Identities

Lets find the six trig functions if

Think where this angle is on the unit circle.

3

2

3

2123

3

2

2

3

3

2

2

1

3

2

Tan

Sin

Cos

3

2

Cos

SinTan

2

3,

2

1

Find the six trig functions of

Think where this angle is on the unit circle.

3

2123

3

2

2

3

3

2

2

1

3

2

Tan

Sin

Cos

3

2

3

3

3

1

3

2

3

32

3

2

3

2

21

2

3

2

Cot

Csc

Sec

How about

4

4

2

2,

2

2

2

2,

2

2

1

2222

4

2

2

4

2

2

4

Tan

Sin

Cos

There are times when Tan or Cot does not exist.

At what angles would this happen?

Ex 1: Find the values of the sine and cosine functions of an angle in standard position with measure θ if the point (3,4) lies on it’s terminal side.     

 Ex 2: If the point (5,12) lies on its terminal side.

Ex 3: Find the sin θ when cos θ = and the terminal side of θ is in the 1st quadrant.       

Ex 4: Find the sin θ when cos θ = and the terminal side of θ is in the 1st quadrant.

7.) The terminal side of an angle θ in standard position contains the point with coordinates (8,-15). Find the value of all six trig functions.        

8.) contains the point (-3,-4)

9.) If csc θ = -2 and θ lies in Quad III, find the values of the five trig functions.       

10.) If sec θ = 2 and θ lies in Quad IV:

YOUR TURN!!!YOUR TURN!!!FILL IN A BLANK UNIT FILL IN A BLANK UNIT

CIRCLE!!CIRCLE!!

YAAAHHHHH!!!YAAAHHHHH!!!