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Trigonometry
The Unit Circle
The Unit Circle
 Imagine a circle on the coordinate 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 xaxis.
 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
y
x
cos    x sin    y
1
1
1
θ
x
y
The Unit Circle
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 .
x
cos    x
1
y
sin    y
1
 The other trigonometric functions can be
defined from these.
Trigonometric Functions
sin   y
1
cosec 
cosecant
y
1
cos  x
1
sec  
x
secant
y
tan  
x
x
cot  
y
cotan
θ
x
y
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
For an angle , in the second
quadrant, the reference
angle is   
In the second quadrant,
sin is positive
cos is negative
tan is negative
Reference angle
Third Quadrant
Original angle θ
For an angle , in the third
quadrant, the reference
angle is  – 
In the third quadrant,
sin is negative
cos is negative
tan is positive
Reference angle
Fourth Quadrant
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
Original angle θ
All Students Take Care
Use the phrase “All Students Take Care”
to remember the signs of the trigometric
functions in the different quadrants.
Students
All
Sine is positive
All positive
Take
Care
Tan is positive
Cos positive
S
A
T
C
Examples
 Find sin240° in surd form.
S
A
T
C
– Draw the angle on the unit circle
– In the 3rd quadrant sine is negative
– Find the angle to nearest x-axis
3
sin 60 
2
60º
3
Page 9 of tables sin 240  
2
Examples
 cosθ = – 0·5. Find the two possible values of θ,
where 0º ≤ θ ≤ 360°.
cosA = 0·5
60º
cos is negative in two quadrants
2nd
180º – 60º
120º
3rd
180º + 60º
240º
S
A
T
C