1. Circular
Trigonometric
Functions

2. Circular Trigonometric FunctionsY
circle…center at (0,0)
radius r…vector with
length/direction r θ X
angle θ… determines
direction

3. Quadrant II Quadrant I 360º Quadrant III Quadrant IV Y-axis 90º r r θ
Terminal side r r θ 0º
X-axis
180º
Initial side
360º
Quadrant III
Quadrant IV
270º

4. Quadrant II Quadrant I Quadrant III Quadrant IV Y-axis -270º -360º
X-axis
-180º
Terminal
side
Initial side 0º r θ
Quadrant III
Quadrant IV
-90º

5. angle θ…measured from positive x-axis,
or initial side, to terminal side
counterclockwise: positive direction
clockwise: negative direction
four quadrants…numbered I, II, III, IV counterclockwise

6. six trigonometric functions for angle θ
whose terminal side passes thru point
(x, y) on circle of radius r
sin θ = y / r csc θ = r / y
cos θ = x / r sec θ = r / x
tan θ = y / x cot θ = x / y
These apply to any angle in any quadrant.

7. For any angle in any quadrant
x2 + y2 = r2 …
So, r is positive by Pythagorean theorem.
(x,y) r y θ x

8. NOTE: right-triangle definitions are special case of circular
functions
when θ is in
quadrant I Y
(x,y) r y θ X x

9. *Reciprocal Identities
sin θ = y / r and csc θ = r / y
cos θ = x / r and sec θ = r / x
tan θ = y / x and cot θ = x / y

10. *Both sets of identities are useful to determine trigonometric
*Ratio Identities
*Both sets of identities are useful
to determine trigonometric
functions of any angle.

11. Students Take Classes Positive trig values in each quadrant: AllY
Students
All
all six positive
sin positive
(csc)
(-, +)
(+, +) II I X
III IV
Take
Classes
(-, -)
(+, -)
tan positive
(cot)
cos positive
(sec)

12. In the ordered pair (x, y), x represents cosine and
REMEMBER:
In the ordered pair (x, y),
x represents cosine and
y represents sine. Y
(-, +)
(+, +) II I X
III IV
(-, -)
(+, -)

13. Examples

14. #1 Draw each angle whose terminal side
passes through the given point, and find
all trigonometric functions of each angle.
θ1: (4, 3)
θ2: (- 4, 3)
θ3: (- 4, -3)
θ4: (4, -3)

15. x =
y =
r = I
(4,3)
sin θ =
cos θ =
tan θ =
csc θ =
sec θ =
cot θ = θ1

16. x = II y = r = (-4,3) θ2 sin θ = cos θ = tan θ = csc θ = sec θ =
cot θ =

17. x = y = r = θ3 (-4,-3) III sin θ = cos θ = tan θ = csc θ = sec θ =
cot θ = θ3
(-4,-3)
III

18. x = y = r = θ4 (4,-3) IV sin θ = cos θ = tan θ = csc θ = sec θ =
cot θ = θ4
(4,-3) IV

19. #2 Given: tan θ = -1 and cos θ is positive:
Draw θ. Show the values for x, y, and r.

20. Given: tan θ = -1 and cos θ is positive:
Find the six trigonometric functions of θ.

21. Calculator
Exercise

22. # 1 Find the value of sin 110º.
(First determine the reference angle.)

23. #2 Find the value of tan 315º.
(First determine the reference angle.)

24. #3 Find the value of cos 230º.
(First determine the reference angle.)

25. Practice

26. #1 Draw the angle whose terminal side passes through the given point .

27. Find all trigonometric functions for angle whose terminal side passes thru .

28. #2 Draw angle: sin θ = 0.6, cos θ is negative.

29. Find all six trigonometric functions:
sin θ = 0.6, cos θ is negative.

30. #3 Find remaining trigonometric functions:
sin θ = , tan θ = 1.000

31. Find remaining trigonometric functions:
sin θ = , tan θ = 1.000