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PRECALCULUS 11
CIRCLE TRIGONOMETRY DEFINITIONS
LINE:
RAY:
ANGLE:
- VERTEX
- INITIAL SIDE
- TERMINAL SIDE
- POSITIVE ANGLE
- NEGATIVE ANGLE
COTERMINAL ANGLES:
- EXAMPLE
ANGLE IN STANDARD POSITION:
- EXAMPLE
RECTANGULAR COORDINATES:
POLAR COORDINATES:
WORKSHEET #1: DRAWING ANGLES IN STANDARD POSITION
A. Sketch the following angles in standard position given the indicated rotation of the
angle.
1. 1
2 of a complete counterclockwise rotation.
2. 1
3 of a complete clockwise rotation.
3. 1
4 of a complete clockwise rotation.
4. 3
2 of a complete counterclockwise rotation.
5. 2
3
8 of a complete clockwise rotation.
6. 1
5
6 of a complete counterclockwise rotation.
B. Label the following Polar Coordinates and state to other possible coterminal angles
(1 positive and 1 negative) that could also be used to locate the same point.
7. P (6;50°) 8. P (4;150°) 9. P (7;-75°) 10. P (3;240°)
11. P (5;-260°) 12. P (2;-450°) 13. P (8;930°) 14. P (6.5;350°)
C. Sketch the following angles in standard position given the indicated equation of its
terminal side (ray). Label a positive and a negative angle.
15. y = 2x, where x≥0 16. y = -3x, where x≤0
17. y = −3
4x, where x≥0 18. y =
5
3x, where x≤0
19. 2x + 7y = 0, where x≤0 20. x = 0, where y≤0
D. Sketch the following angles in standard position given the following points contained
within its terminal side. Then state the equation of its terminal side. (Ray rotation of the
angle.
21. P (-2, 3) 22. P (4, 6) 23. P (3,-1) 24. P (0,-6)
25. P (-5,-4) 26. P (-2, -2) 27. P (-8, 0) 28. P (7, -4)
E. Why does Stan the Ski Jumper who performs a “1260” in international competitions
need special skis?
TRIGONOMETRIC RATIOS FOR ANGLES IN STANDARD POSITION PART 1
For each angle estimate x and y to the nearest tenth. Then calculate sin, cos and tan to nearest
hundredth.
θ 0° 30° 45° 60° 90° 120° 135° 150° 180°
X
Y
r
cos θ
sin θ
tan θ
θ 180° 210° 225° 240° 270° 300° 315° 330° 360°
X
Y
r
cos θ
sin θ
tan θ
CUT OUT PROTRACTOR
RECIPROCAL TRIGONOMETRIC FUNCTIONS, QUADRANTS AND SIGNS
A. For the following state the value of the reciprocal trigonometric ratio given its
corresponding primary ratio.
1. sin θ = 2
7 , csc θ =____
2. cos θ = −
3
4 , sec θ =____
3. sec θ = −
5
2 , cos θ =____
4. cot θ = 5 , tan θ =____
5. tan θ = −
4
3 , cot θ =____
6. csc θ = 2 5
5 , sin θ =____
7. cos θ = 2
5 , sin θ = − 21
5 ,
cot θ =____
8. cos θ = 2 5
5 , sin θ = − 5
5 ,
cot θ =____
B. Name the quadrants in which the terminal side of θ may lie.
9. cos θ > 0
10.
sin θ < 0 11.
tan θ > 0
12. cot θ < 0
13.
sec θ < 0 14.
csc θ > 0
15. cos θ < 0, csc θ < 0
16.
sin θ < 0, sec θ < 0 17.
sin θ < 0, tan θ < 0
18. cos θ < 0, tan θ > 0
19. sec θ < 0, cot θ > 0
20. tan θ < 0, csc θ > 0
C. State the Domain and Range for the following functions.
TRIGONOMETRIC FUNCTION DOMAIN RANGE
1. F(x) = cos(x)
2. F(x) = sin(x)
3. F(x) = tan(x)
4. F(x) = sec(x)
5. F(x) = csc(x)
6. F(x) = cot(x)
THE 6 TRIGONOMETRIC RATIOS
A. State the 6 Trigonometric values for θ, given the following points located on its
terminal side.
1. P (8, 15) 2. P (-4, 3) 3. P (5,-12) 4. P (-8,-6)
5. P (-1,-1) 6. P (-2, 4) 7. P ( 3 , -1) 8. P (0, -4)
B. State the 6 Trigonometric values for θ, given the equation of its terminal side.
15. y = -2x, where x≥0
16. y + x = 0, where y≤0
17. y = −
3
4x, where x≥0
18. y =
5
3x, where x≤0
19. 2x - 3y = 0, where x≤0 20. y = 0, where x≤0
C. State the other 5 Trigonometric values for θ, given one of its trigonometric ratios.
21. cos θ = −1
2 , csc θ < 0
22. sin θ = −1
2 , sec θ > 0
23. sec θ = −5
3 , tan θ > 0
24.
cot θ = -6 , cos θ < 0
25. tan θ = 1 , sin θ < 0
26. sec θ = 2 , cot θ > 0
27. csc θ = 2 , tan θ = > 0 28. cot θ = − 3 , sin θ > 0
THE 6 TRIGONOMETRIC RATIOS ANSWER SHEET:
1 2 3 4 5 6 7
X
Y
r
cos θ
sin θ
tan θ
sec θ
csc θ
cot θ 8 9 10 11 12 13 14
X
Y
r
cos θ
sin θ
tan θ
sec θ
csc θ
cot θ 15 16 17 18 19 20 21
X
Y
r
cos θ
sin θ
tan θ
sec θ
csc θ
cot θ 22 23 24 25 26 27 28
X
Y
r
cos θ
sin θ
tan θ
sec θ
csc θ
cot θ
REFERENCE ANGLES
A. State the reference angle for θ.
1. 135° 2. 300° 3. 280° 4. 172°
5. -115° 6. -35° 7. 440° 8. 590°
9. 1200° 10. 1150° 11. -850° 12. -865°
B. Express the following as a function of a positive acute angle.
13. cos132° 14. sin165° 15. cot230° 16. tan295°
17. sec(-305°) 18. sin(-655°) 19. cos(-830)° 20. tan893°
21. cot968° 22. sec(-239°) 23. csc(-102°) 24. csc(-400°)
C. Find the angle in standard position θ to one decimal place for each of the following.
Where 0° ≤ θ ≤ 360°
21. cos θ = −
1
2 , tan θ < 0
22. sin θ = .2306 , cos θ < 0
23. csc θ = −
5
3 , tan θ > 0
24. cot θ = -1.276 , cos θ > 0
25. tan θ = -1.393 , sin θ > 0
26. sec θ = − 2 , cot θ > 0
27. csc θ = 2.419 , tan θ < 0 28. cot θ = − 3 , sin θ > 0
D. Find the corresponding RECTANGULAR COORDINATES for.
29. P (6;144°) 30. P (11;284°) 31. P (7;-75°) 32. P (3;243°)
D. Find the corresponding POLAR COORDINATES for.
33. P (-3, 4) 34. P (-12,-5) 35. P (4,-6) 36. P (1,-7)
SPECIAL ANGLES AND THEIR EXACT VALUES
FILL IN THE FOLLOWING TABLE (MUST MEMORIZE)
θ x y
0°
30°
45°
60°
90°
A. Evaluate the following using EXACT VALUES.
1. sec tan2 260 60°− °
2. csc cot2 2315 315°− °
3. sin120° + cos210° + tan300°
4. tan225° + cot315° - sec150°
5. sin120° cos150° + cos120° sin150°
6. cos135° cos45° + sin135° sin45°
7. sin330° cos120° tan135°
8. (cos330° + sin60°) (tan30° + cot240°)
9. (tan225° + sin270°) cos150°
PROVE THE FOLLOWING
10. sin60° = 2sin30° cos30°
11. cos90° = 2cos245° – tan
245°
12. sin30° = 2
60cos1 °−
13. tan30° = °
°−
60sin
60cos1
14. cos240° = cos2120° – sin
2120°
15. tan210° = °°−
°+°
150tan60tan1
150tan60tan
TRIGONOMETRIC EQUATIONS SOLVING
Solve for θ to one decimal place or EXACT VALUES for each of the
following. Where 0° ≤ θ ≤ 360°
1. 2sinθ = 1
2. 2cosθ + 1 = 3cosθ + 2
3. tan2θ - tanθ = 0
4. 2sin2θ + sinθ - 1 = 0
5. (cosθ + 1) (2sin2θ - 1) = 0
6. 2cscθ + 1 = 0
7. cosθ (cosθ - 1) = 0
8. tanθ cosθ + cosθ = 0
9. 2 sec2θ - 4 = 0
10. 4 sin2θ = 3
11. 2sin2θ - 5sinθ - 3 = 0
12. 3cos2θ - 3cosθ = 0
13. sinθ - cosθ = 0
14. 7sinθ + 4 = 4sinθ - 1
15. 2cotθ - 11 = -3cotθ - 17
16. 4cos2θ = 1
17. 3tan2θ - 2tanθ = 4
18. 4cos2θ + 5cosθ = 6
19. 4 sin22θ = 3
20. 8cos22θ + 2cos2θ - 1 = 0
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