Download Geometry H Additional Topics in Trigonometry 2. Finding Quadrant

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 200° 36. cos 568° 37. tan 704°
38. sin(-80°) 39. cos(-170°) 40. tan(-690°)