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Geometry H
Additional Topics in Trigonometry
2. Finding Quadrant of Terminal Ray
Finding Sign of Trigonometric Functions
Finding Co-terminal Angles
Finding quadrant of a terminal ray
The degree measure of an angle tells us
how far – and in which direction – the
terminal ray moves.
Positive movement
If 0 < θ < 90, θ terminates in I.
If 90 < θ < 180, θ terminates in II.
If 180 < θ < 270, θ terminates in III.
If 270 < θ < 360, θ terminates in IV.
Negative movement
If -90 < θ < 0, θ terminates in IV.
If -180 < θ < -90, θ terminates in III.
If -270 < θ < -180, θ terminates in II.
If -360 < θ < -270, θ terminates in I.
If θ is a multiple of 90 (0, 90,
180, 270, 360), it terminates
between two quadrants. An angle which
does not terminate in any quadrant is
called a quadrant angle.
Examples
A 76 angle terminates in Quad I.
A 312 angle terminates in Quad IV.
A -100 angle terminates in Quad III.
A -210 angle terminates in Quad II.
A -180 angle is a quadrant angle.
If θ terminates in Quad III or Quad IV,
sin θ is negative since sin θ = y/r
and y is negative in Quad III and IV.
If θ terminates in Quad I or Quad IV,
cos θ is positive since cos θ = x/r
and x is positive in Quad I and IV.
If θ terminates in Quad II or Quad III,
cos θ is negative since cos θ = x/r
and x is negative in Quad II and III.
If θ terminates in Quad I or Quad III,
tan θ is positive since tan θ = y/x
and y/x is positive in Quad I and III.
Finding the sign of functions
Since the radius, r, is always positive,
the sign of a trig function depends on the
sign of x and y (positive, negative, or 0).
If θ terminates in Quad I or Quad II,
sin θ is positive since sin θ = y/r
and y is positive in Quad I and II.
If θ terminates in Quad II or Quad IV,
tan θ is negative since tan θ = y/x
and y/x is negative in Quad II and IV.
Examples
Sin 100 is positive because a 100 angle
terminates in Quad II and y is positive
there.
Geometry H
Additional Topics in Trigonometry
2. Finding Quadrant of Terminal Ray
Finding Sign of Trigonometric Functions
Finding Co-terminal Angles
Cos -40 is positive because a -40 angle
terminates in Quad IV and x is positive
there.
Tan 136 is negative because a 136
angle terminates in Quad II and y/x is
negative there.
Tan(-320) is positive because
tan (-320) = tan 40. A 40 angle
terminates in Quad I, and tangent is
positive in quad I.
Exercises
Finding co-terminal angles
The angle terminates in what quadrant?
Any angle is co-terminal with any other
angle from which it differs by 360 - or
any integral multiple of 360.
1. 110
2. 205
3. 350
4. 180
5. -20
6. -200
7. -90
8. -290
If two angles are co-terminal, they have
the same sine, the same cosine, and the
same tangent. They are essentially two
names for the same angle.
9. 470 10. 750
So, if θ is not between 0 and 360, you
can transform it into one. A 410 can be
expressed as a 50 (410 - 360) angle.
A -120 angle can be expressed as a
240 (-120 + 360) angle.
Is the quantity as positive or negative ?
Examples
740 is co-terminal with 20.
(740-360 = 380
and 380-360 = 20)
-200 is co-terminal with 160.
(-200 + 360 = 160.)
sin 500 = sin 140
(500 – 360 = 140)
Cos(-420) = cos 300
(-420 + 360 + 360 = 300)
Tan(-150) = tan 210
(-150 + 360 = 210)
Sin 600 is negative because
sin 600 = sin 240.
240 terminates in Quad III, and sine
is negative in Quad IV.
23. sin 456 24. cos 800
11. 532
12. 900
13. -50 14. -300 15. -360 16. -400
17. sin 48
18. cos 190
19. tan 318
20. sin(-99) 21. cos(-315) 22. tan(-202)
25. tan 714
26. sin(-390) 27. cos(-600) 28. tan(-850)
Express in terms of an angle between 0
and 360.
29. 680
30. 457
31. 1200
32. -235
33. -526
34. -1010
35. sin 400 36. cos 568 37. tan 704
38. sin(-80) 39. cos(-170) 40. tan(-690)