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