Trigonometry

The Unit Circle

The Unit Circle

Imagine a circle on the co-ordinate plane, with its center at the origin, and a radius of 1.

Choose a point on the circle somewhere in the first quadrant.

The Unit Circle

Connect the origin to the point, and from that point drop a perpendicular to the x-axis.

This creates a right triangle with hypotenuse of 1.

1

The Unit Circle The length of sides of the triangle are

the x and y co-ordinates of the chosen point.

Applying the definitions of

the trigonometric ratios to

this triangle gives

cos1

xx

x

y1

θ

sin1

yy

The co-ordinates of the chosen point are the cosine and sine of the angle .

This provides a way to define functions sin and cos for all real numbers .

The other trigonometric functions can be defined from these.

The Unit Circle

sin1

yy cos

1

xx

Trigonometric Functions

sin y

cos x

tany

x

1cosec

y

1sec

x

cotx

y

x

y1

θ

cosecant

secant

cotan

Around the Circle

As that point moves around the unit circle into the second, third and fourth quadrants, the new definitions of the trigonometric functions still hold.

Reference Angles

The angles whose terminal sides fall in the 2nd, 3rd, and 4th quadrants will have values of sine, cosine and other trig functions which are identical (except for sign) to the values of angles in 1st quadrant.

The acute angle which produces the same values is called the reference angle.

Second Quadrant

Original angle θ

Reference angle

For an angle , in the second quadrant, the reference angle is

In the second quadrant,

sin is positive

cos is negative

tan is negative

Third Quadrant

Original angle θ

Reference angle

For an angle , in the third quadrant, the reference angle is –

In the third quadrant,

sin is negative

cos is negative

tan is positive

Fourth Quadrant

Original angle θ

Reference angle

For an angle , in the fourth quadrant, the reference angle is 2

In the fourth quadrant,

sin is negative

cos is positive

tan is negative

All Students Take Care

AllStudents

Take Care

Use the phrase “All Students Take Care” to remember the signs of the trigometric functions in the different quadrants.

CS A

T Cine is pS ositive ll posA itive

os posC itivean is pT ositive

Examples

Find sin240° in surd form.CS A

T C– Draw the angle on the unit circle

– In the 3rd quadrant sine is negative

– Find the angle to nearest x-axis 60º

3sin 240

2 Page 9 of tables

3sin 60

2

Examples

cosθ = – 0·5. Find the two possible values of θ, where 0º ≤ θ ≤ 360°.

S A

T C

60ºcosA = 0·5

cos is negative in two quadrants

2nd

3rd 180º + 60º 240º

180º – 60º 120º