1. Trigonometry
The Unit Circle

2. 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.

3. 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

4. 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 1 y θ x

5. 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 .
 The other trigonometric functions can be defined from these.

6. Trigonometric Functions
cosecant 1 y
secant θ x
cotan

7. 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.

8. 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.

9. Second Quadrant
For an angle , in the second quadrant, the reference angle is   
In the second quadrant,
sin is positive
cos is negative
tan is negative
Original angle θ
Reference angle

10. Third Quadrant
For an angle , in the third quadrant, the reference angle is  – 
In the third quadrant,
sin is negative
cos is negative
tan is positive
Original angle θ
Reference angle

11. Fourth Quadrant
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
Reference angle
Original angle θ

12. All Students Take Care Students All Take Care
Use the phrase “All Students Take Care” to remember the signs of the trigometric functions in the different quadrants. C
S A
T C
Students
All
Take
Care

13. Examples  Find sin240° in surd form. C 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
60º
Page 9 of tables

14. Examples  cosθ = – 0·5. Find the two possible values of θ,
where 0º ≤ θ ≤ 360°.
60º
cosA = 0·5
S A
T C
cos is negative in two quadrants
2nd
3rd
180º – 60º
120º
180º + 60º
240º