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5.E: Trigonometric Functions (Exercises) - Mathematics


5.1: Angles

In this section, we will examine properties of angles.

Verbal

1) Draw an angle in standard position. Label the vertex, initial side, and terminal side.

Answer

2) Explain why there are an infinite number of angles that are coterminal to a certain angle.

3) State what a positive or negative angle signifies, and explain how to draw each.

Answer

Whether the angle is positive or negative determines the direction. A positive angle is drawn in the counterclockwise direction, and a negative angle is drawn in the clockwise direction.

4) How does radian measure of an angle compare to the degree measure? Include an explanation of (1) radian in your paragraph.

5) Explain the differences between linear speed and angular speed when describing motion along a circular path.

Answer

Linear speed is a measurement found by calculating distance of an arc compared to time. Angular speed is a measurement found by calculating the angle of an arc compared to time.

Graphical

For the exercises 6-21, draw an angle in standard position with the given measure.

6) (30^{circ})

7) (300^{circ})

Answer

8) (-80^{circ})

9) (135^{circ})

Answer

10) (-150^{circ})

11) (dfrac{2π}{3})

Answer

12) (dfrac{7π}{4})

13) (dfrac{5π}{6})

Answer

14) (dfrac{π}{2})

15) (−dfrac{π}{10})

Answer

16) (415^{circ})

17) (-120^{circ})

Answer

(240^{circ})

18) (-315^{circ})

19)(dfrac{22π}{3})

Answer

(dfrac{4π}{3})

20) (−dfrac{π}{6})

21) (−dfrac{4π}{3})

Answer

(dfrac{2π}{3})

For the exercises 22-23, refer to Figure below. Round to two decimal places.

22) Find the arc length.

23) Find the area of the sector.

Answer

(dfrac{27π}{2}≈11.00 ext{ in}^2)

For the exercises 24-25, refer to Figure below. Round to two decimal places.

24) Find the arc length.

25) Find the area of the sector.

Answer

(dfrac{81π}{20}≈12.72 ext{ cm}^2)

Algebraic

For the exercises 26-32, convert angles in radians to degrees.

26) (dfrac{3π}{4}) radians

27) (dfrac{π}{9}) radians

Answer

(20^{circ})

28) (−dfrac{5π}{4}) radians

29) (dfrac{π}{3}) radians

Answer

(60^{circ})

30) (−dfrac{7π}{3}) radians

31) (−dfrac{5π}{12}) radians

Answer

(-75^{circ})

32) (dfrac{11π}{6}) radians

For the exercises 33-39, convert angles in degrees to radians.

33) (90^{circ})

Answer

(dfrac{π}{2}) radians

34) (100^{circ})

35) (-540^{circ})

Answer

(−3π) radians

36) (-120^{circ})

37) (180^{circ})

Answer

(π) radians

38) (-315^{circ})

39) (150^{circ})

Answer

(dfrac{5π}{6}) radians

For the exercises 40-45, use to given information to find the length of a circular arc. Round to two decimal places.

40) Find the length of the arc of a circle of radius (12) inches subtended by a central angle of (dfrac{π}{4}) radians.

41) Find the length of the arc of a circle of radius (5.02) miles subtended by the central angle of (dfrac{π}{3}).

Answer

(dfrac{5.02π}{3}≈5.26) miles

42) Find the length of the arc of a circle of diameter (14) meters subtended by the central angle of (dfrac{5pi }{6}).

43) Find the length of the arc of a circle of radius (10) centimeters subtended by the central angle of (50^{circ}).

Answer

(dfrac{25π}{9}≈8.73) centimeters

44) Find the length of the arc of a circle of radius (5) inches subtended by the central angle of (220^{circ}).

45) Find the length of the arc of a circle of diameter (12) meters subtended by the central angle is (63^{circ}).

Answer

(dfrac{21π}{10}≈6.60) meters

For the exercises 46-49, use the given information to find the area of the sector. Round to four decimal places.

46) A sector of a circle has a central angle of (45^{circ}) and a radius (6) cm.

47) A sector of a circle has a central angle of (30^{circ}) and a radius of (20) cm.

Answer

(104.7198; cm^2)

48) A sector of a circle with diameter (10) feet and an angle of (dfrac{π}{2}) radians.

49) A sector of a circle with radius of (0.7) inches and an angle of (π) radians.

Answer

(0.7697; in^2)

For the exercises 50-53, find the angle between (0^{circ}) and (360^{circ}) that is coterminal to the given angle.

50) (-40^{circ})

51) (-110^{circ})

Answer

(250^{circ})

52) (700^{circ})

53) (1400^{circ})

Answer

(320^{circ})

For the exercises 54-57, find the angle between (0) and (2pi ) in radians that is coterminal to the given angle.

54) (−dfrac{π}{9})

55) (dfrac{10π}{3})

Answer

(dfrac{4π}{3})

56) (dfrac{13π}{6})

57) (dfrac{44π}{9})

Answer

(dfrac{8π}{9})

Real-World Applications

58) A truck with (32)-inch diameter wheels is traveling at (60) mi/h. Find the angular speed of the wheels in rad/min. How many revolutions per minute do the wheels make?

59) A bicycle with (24)-inch diameter wheels is traveling at (15) mi/h. How many revolutions per minute do the wheels make?

Answer

(1320) rad (210.085) RPM

60) A wheel of radius (8) inches is rotating (15^{circ}/s). What is the linear speed (v), the angular speed in RPM, and the angular speed in rad/s?

61) A wheel of radius (14) inches is rotating (0.5 ext{rad/s}). What is the linear speed (v), the angular speed in RPM, and the angular speed in deg/s?

Answer

(7) in./s, (4.77) RPM, (28.65) deg/s

62) A CD has diameter of (120) millimeters. When playing audio, the angular speed varies to keep the linear speed constant where the disc is being read. When reading along the outer edge of the disc, the angular speed is about (200) RPM (revolutions per minute). Find the linear speed.

63) When being burned in a writable CD-R drive, the angular speed of a CD is often much faster than when playing audio, but the angular speed still varies to keep the linear speed constant where the disc is being written. When writing along the outer edge of the disc, the angular speed of one drive is about (4800) RPM (revolutions per minute). Find the linear speed if the CD has diameter of (120) millimeters.

Answer

(1,809,557.37 ext{ mm/min}=30.16 ext{ m/s})

64) A person is standing on the equator of Earth (radius (3960) miles). What are his linear and angular speeds?

65) Find the distance along an arc on the surface of Earth that subtends a central angle of (5) minutes ((1 ext{ minute}=dfrac{1}{60} ext{ degree})). The radius of Earth is (3960) miles.

Answer

(5.76) miles

66) Find the distance along an arc on the surface of Earth that subtends a central angle of (7) minutes ((1 ext{ minute}=dfrac{1}{60} ext{ degree})). The radius of Earth is (3960) miles.

67) Consider a clock with an hour hand and minute hand. What is the measure of the angle the minute hand traces in (20) minutes?

Answer

(120°)

Extensions

68) Two cities have the same longitude. The latitude of city A is (9.00) degrees north and the latitude of city B is (30.00) degree north. Assume the radius of the earth is (3960) miles. Find the distance between the two cities.

69) A city is located at (40) degrees north latitude. Assume the radius of the earth is (3960) miles and the earth rotates once every (24) hours. Find the linear speed of a person who resides in this city.

Answer

(794) miles per hour

70) A city is located at (75) degrees north latitude. Find the linear speed of a person who resides in this city.

71) Find the linear speed of the moon if the average distance between the earth and moon is (239,000) miles, assuming the orbit of the moon is circular and requires about (28) days. Express answer in miles per hour.

Answer

(2,234) miles per hour

72) A bicycle has wheels (28) inches in diameter. A tachometer determines that the wheels are rotating at (180) RPM (revolutions per minute). Find the speed the bicycle is traveling down the road.

73) A car travels (3) miles. Its tires make (2640) revolutions. What is the radius of a tire in inches?

Answer

(11.5) inches

74) A wheel on a tractor has a (24)-inch diameter. How many revolutions does the wheel make if the tractor travels (4) miles?

5.2: Unit Circle - Sine and Cosine Functions

Verbal

1) Describe the unit circle.

Answer

The unit circle is a circle of radius (1) centered at the origin.

2) What do the (x)- and (y)-coordinates of the points on the unit circle represent?

3) Discuss the difference between a coterminal angle and a reference angle.

Answer

Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle, (t), formed by the terminal side of the angle (t) and the horizontal axis.

4) Explain how the cosine of an angle in the second quadrant differs from the cosine of its reference angle in the unit circle.

5) Explain how the sine of an angle in the second quadrant differs from the sine of its reference angle in the unit circle.

Answer

The sine values are equal.

Algebraic

For the exercises 6-9, use the given sign of the sine and cosine functions to find the quadrant in which the terminal point determined by (t) lies.

6) ( sin (t)<0) and ( cos (t)<0)

7) ( sin (t)>0) and ( cos (t)>0)

Answer

( extrm{I})

8) ( sin (t)>0 ) and ( cos (t)<0)

9) ( sin (t)<0 ) and ( cos (t)>0)

Answer

( extrm{IV})

For the exercises 10-22, find the exact value of each trigonometric function.

10) (sin dfrac{π}{2})

11) (sin dfrac{π}{3})

Answer

(dfrac{sqrt{3}}{2})

12) ( cos dfrac{π}{2})

13) ( cos dfrac{π}{3})

Answer

(dfrac{1}{2})

14) ( sin dfrac{π}{4})

15) ( cos dfrac{π}{4})

Answer

(dfrac{sqrt{2}}{2})

16) ( sin dfrac{π}{6})

17) ( sin π)

Answer

(0)

18) ( sin dfrac{3π}{2})

19) ( cos π)

Answer

(−1)

20) ( cos 0)

21) (cos dfrac{π}{6})

Answer

(dfrac{sqrt{3}}{2})

22) ( sin 0)

Numeric

For the exercises 23-33, state the reference angle for the given angle.

23) (240°)

Answer

(60°)

24) (−170°)

25) (100°)

Answer

(80°)

26) (−315°)

27) (135°)

Answer

(45°)

28) (dfrac{5π}{4})

29) (dfrac{2π}{3})

Answer

(dfrac{π}{3})

30) (dfrac{5π}{6})

31) (−dfrac{11π}{3})

Answer

(dfrac{π}{3})

32) (dfrac{−7π}{4})

33) (dfrac{−π}{8})

Answer

(dfrac{π}{8})

For the exercises 34-49, find the reference angle, the quadrant of the terminal side, and the sine and cosine of each angle. If the angle is not one of the angles on the unit circle, use a calculator and round to three decimal places.

34) (225°)

35) (300°)

Answer

(60°), Quadrant IV, ( sin (300°)=−dfrac{sqrt{3}}{2}, cos (300°)=dfrac{1}{2})

36) (320°)

37) (135°)

Answer

(45°), Quadrant II, ( sin (135°)=dfrac{sqrt{2}}{2}, cos (135°)=−dfrac{sqrt{2}}{2})

38) (210°)

39) (120°)

Answer

(60°), Quadrant II, (sin (120°)=dfrac{sqrt{3}}{2}), (cos (120°)=−dfrac{1}{2})

40) (250°)

41) (150°)

Answer

(30°), Quadrant II, ( sin (150°)=frac{1}{2}), (cos(150°)=−dfrac{sqrt{3}}{2})

42) (dfrac{5π}{4})

43) (dfrac{7π}{6})

Answer

(dfrac{π}{6}), Quadrant III, (sin left( dfrac{7π}{6} ight )=−dfrac{1}{2}), (cos left (dfrac{7π}{6} ight)=−dfrac{sqrt{3}}{2})

44) (dfrac{5π}{3})

45) (dfrac{3π}{4})

Answer

(dfrac{π}{4}), Quadrant II, (sin left(dfrac{3π}{4} ight)=dfrac{sqrt{2}}{2}), (cosleft(dfrac{4π}{3} ight)=−dfrac{sqrt{2}}{2})

46) (dfrac{4π}{3})

47) (dfrac{2π}{3})

Answer

(dfrac{π}{3}), Quadrant II, ( sin left(dfrac{2π}{3} ight)=dfrac{sqrt{3}}{2}), ( cos left(dfrac{2π}{3} ight)=−dfrac{1}{2})

48) (dfrac{5π}{6})

49) (dfrac{7π}{4})

Answer

(dfrac{π}{4}), Quadrant IV, ( sin left(dfrac{7π}{4} ight)=−dfrac{sqrt{2}}{2}), ( cos left(dfrac{7π}{4} ight)=dfrac{sqrt{2}}{2})

For the exercises 50-59, find the requested value.

50) If (cos (t)=dfrac{1}{7}) and (t) is in the (4^{th}) quadrant, find ( sin (t)).

51) If ( cos (t)=dfrac{2}{9}) and (t) is in the (1^{st}) quadrant, find (sin (t)).

Answer

(dfrac{sqrt{77}}{9})

52) If (sin (t)=dfrac{3}{8}) and (t) is in the (2^{nd}) quadrant, find ( cos (t)).

53) If ( sin (t)=−dfrac{1}{4}) and (t) is in the (3^{rd}) quadrant, find (cos (t)).

Answer

(−dfrac{sqrt{15}}{4})

54) Find the coordinates of the point on a circle with radius (15) corresponding to an angle of (220°).

55) Find the coordinates of the point on a circle with radius (20) corresponding to an angle of (120°).

Answer

((−10,10sqrt{3}))

56) Find the coordinates of the point on a circle with radius (8) corresponding to an angle of (dfrac{7π}{4}).

57) Find the coordinates of the point on a circle with radius (16) corresponding to an angle of (dfrac{5π}{9}).

Answer

((–2.778,15.757))

58) State the domain of the sine and cosine functions.

59) State the range of the sine and cosine functions.

Answer

([–1,1])

Graphical

For the exercises 60-79, use the given point on the unit circle to find the value of the sine and cosine of (t).

60)

61)

Answer

( sin t=dfrac{1}{2}, cos t=−dfrac{sqrt{3}}{2})

62)

63)

Answer

( sin t=− dfrac{sqrt{2}}{2}, cos t=−dfrac{sqrt{2}}{2})

64)

65)

Answer

( sin t=dfrac{sqrt{3}}{2},cos t=−dfrac{1}{2})

66)

67)

Answer

( sin t=− dfrac{sqrt{2}}{2}, cos t=dfrac{sqrt{2}}{2})

68)

69)

Answer

( sin t=0, cos t=−1)

70)

71)

Answer

( sin t=−0.596, cos t=0.803)

72)

73)

Answer

(sin t=dfrac{1}{2}, cos t= dfrac{sqrt{3}}{2})

74)

75)

Answer

( sin t=−dfrac{1}{2}, cos t= dfrac{sqrt{3}}{2} )

76)

77)

Answer

( sin t=0.761, cos t=−0.649 )

78)

79)

Answer

( sin t=1, cos t=0)

Technology

For the exercises 80-89, use a graphing calculator to evaluate.

80) ( sin dfrac{5π}{9})

81) (cos dfrac{5π}{9})

Answer

(−0.1736)

82) ( sin dfrac{π}{10})

83) ( cos dfrac{π}{10})

Answer

(0.9511)

84) ( sin dfrac{3π}{4})

85) (cos dfrac{3π}{4})

Answer

(−0.7071)

86) ( sin 98° )

87) ( cos 98° )

Answer

(−0.1392)

88) ( cos 310° )

89) ( sin 310° )

Answer

(−0.7660)

Extensions

For the exercises 90-99, evaluate.

90) ( sin left(dfrac{11π}{3} ight) cos left(dfrac{−5π}{6} ight))

91) ( sin left(dfrac{3π}{4} ight) cos left(dfrac{5π}{3} ight) )

Answer

(dfrac{sqrt{2}}{4})

92) ( sin left(− dfrac{4π}{3} ight) cos left(dfrac{π}{2} ight))

93) ( sin left(dfrac{−9π}{4} ight) cos left(dfrac{−π}{6} ight))

Answer

(−dfrac{sqrt{6}}{4})

94) ( sin left(dfrac{π}{6} ight) cos left(dfrac{−π}{3} ight) )

95) ( sin left(dfrac{7π}{4} ight) cos left(dfrac{−2π}{3} ight) )

Answer

(dfrac{sqrt{2}}{4})

96) ( cos left(dfrac{5π}{6} ight) cos left(dfrac{2π}{3} ight))

97) ( cos left(dfrac{−π}{3} ight) cos left(dfrac{π}{4} ight) )

Answer

(dfrac{sqrt{2}}{4})

98) ( sin left(dfrac{−5π}{4} ight) sin left(dfrac{11π}{6} ight))

99) ( sin (π) sin left(dfrac{π}{6} ight) )

Answer

(0)

Real-World Applications

For the exercises 100-104, use this scenario: A child enters a carousel that takes one minute to revolve once around. The child enters at the point ((0,1)), that is, on the due north position. Assume the carousel revolves counter clockwise.

100) What are the coordinates of the child after (45) seconds?

101) What are the coordinates of the child after (90) seconds?

Answer

((0,–1))

102) What is the coordinates of the child after (125) seconds?

103) When will the child have coordinates ((0.707,–0.707)) if the ride lasts (6) minutes? (There are multiple answers.)

Answer

(37.5) seconds, (97.5) seconds, (157.5) seconds, (217.5) seconds, (277.5) seconds, (337.5) seconds

104) When will the child have coordinates ((−0.866,−0.5)) if the ride last (6) minutes?

5.3: The Other Trigonometric Functions

Verbal

1) On an interval of ([ 0,2π )), can the sine and cosine values of a radian measure ever be equal? If so, where?

Answer

Yes, when the reference angle is (dfrac{π}{4}) and the terminal side of the angle is in quadrants I and III. Thus, at (x=dfrac{π}{4},dfrac{5π}{4}), the sine and cosine values are equal.

2) What would you estimate the cosine of (pi ) degrees to be? Explain your reasoning.

3) For any angle in quadrant II, if you knew the sine of the angle, how could you determine the cosine of the angle?

Answer

Substitute the sine of the angle in for (y) in the Pythagorean Theorem (x^2+y^2=1). Solve for (x) and take the negative solution.

4) Describe the secant function.

5) Tangent and cotangent have a period of (π). What does this tell us about the output of these functions?

Answer

The outputs of tangent and cotangent will repeat every (π) units.

Algebraic

For the exercises 6-17, find the exact value of each expression.

6) ( an dfrac{π}{6})

7) (sec dfrac{π}{6})

Answer

(dfrac{2sqrt{3}}{3})

8) ( csc dfrac{π}{6})

9) ( cot dfrac{π}{6})

Answer

(sqrt{3})

10) ( an dfrac{π}{4})

11) ( sec dfrac{π}{4})

Answer

(sqrt{2})

12) ( csc dfrac{π}{4})

13) ( cot dfrac{π}{4})

Answer

(1)

14) ( an dfrac{π}{3})

15) ( sec dfrac{π}{3})

Answer

(2)

16) ( csc dfrac{π}{3})

17) ( cot dfrac{π}{3})

Answer

(dfrac{sqrt{3}}{3})

For the exercises 18-48, use reference angles to evaluate the expression.

18) ( an dfrac{5π}{6})

19) ( sec dfrac{7π}{6})

Answer

(−dfrac{2sqrt{3}}{3})

20) ( csc dfrac{11π}{6})

21) ( cot dfrac{13π}{6})

Answer

(sqrt{3})

22) ( an dfrac{7π}{4})

23) ( sec dfrac{3π}{4})

Answer

(−sqrt{2})

24) ( csc dfrac{5π}{4})

25) ( cot dfrac{11π}{4})

Answer

(−1)

26) ( an dfrac{8π}{3})

27) ( sec dfrac{4π}{3})

Answer

(−2)

28) ( csc dfrac{2π}{3})

29) ( cot dfrac{5π}{3})

Answer

(−dfrac{sqrt{3}}{3})

30) ( an 225°)

31) ( sec 300°)

Answer

(2)

32) ( csc 150°)

33) ( cot 240°)

Answer

(dfrac{sqrt{3}}{3})

34) ( an 330°)

35) ( sec 120°)

Answer

(−2)

36) ( csc 210°)

37) ( cot 315°)

Answer

(−1)

38) If ( sin t= dfrac{3}{4}), and (t) is in quadrant II, find ( cos t, sec t, csc t, an t, cot t ).

39) If ( cos t=−dfrac{1}{3},) and (t) is in quadrant III, find ( sin t, sec t, csc t, an t, cot t).

Answer

If (sin t=−dfrac{2sqrt{2}}{3}, sec t=−3, csc t=−csc t=−dfrac{3sqrt{2}}{4}, an t=2sqrt{2}, cot t= dfrac{sqrt{2}}{4})

40) If ( an t=dfrac{12}{5},) and (0≤t< dfrac{π}{2}), find ( sin t, cos t, sec t, csc t,) and (cot t).

41) If ( sin t= dfrac{sqrt{3}}{2}) and ( cos t=dfrac{1}{2},) find ( sec t, csc t, an t,) and ( cot t).

Answer

( sec t=2, csc t=csc t=dfrac{2sqrt{3}}{3}, an t= sqrt{3}, cot t= dfrac{sqrt{3}}{3})

42) If ( sin 40°≈0.643 ; cos 40°≈0.766 ; sec 40°,csc 40°, an 40°, ext{ and } cot 40°).

43) If ( sin t= dfrac{sqrt{2}}{2},) what is the ( sin (−t))?

Answer

(−dfrac{sqrt{2}}{2})

44) If ( cos t= dfrac{1}{2},) what is the ( cos (−t))?

45) If ( sec t=3.1,) what is the ( sec (−t))?

Answer

(3.1)

46) If ( csc t=0.34,) what is the ( csc (−t))?

47) If ( an t=−1.4,) what is the ( an (−t))?

Answer

(1.4)

48) If ( cot t=9.23,) what is the ( cot (−t))?

Graphical

For the exercises 49-51, use the angle in the unit circle to find the value of the each of the six trigonometric functions.

49)

Answer

( sin t= dfrac{sqrt{2}}{2}, cos t= dfrac{sqrt{2}}{2}, an t=1,cot t=1,sec t= sqrt{2}, csc t= csc t= sqrt{2} )

50)

51)

Answer

( sin t=−dfrac{sqrt{3}}{2}, cos t=−dfrac{1}{2}, an t=sqrt{3}, cot t= dfrac{sqrt{3}}{3}, sec t=−2, csc t=−csc t=−dfrac{2sqrt{3}}{3} )

Technology

For the exercises 52-61, use a graphing calculator to evaluate.

52) ( csc dfrac{5π}{9})

53) ( cot dfrac{4π}{7})

Answer

(–0.228)

54) ( sec dfrac{π}{10})

55) ( an dfrac{5π}{8})

Answer

(–2.414)

56) ( sec dfrac{3π}{4})

57) ( csc dfrac{π}{4})

Answer

(1.414)

58) ( an 98°)

59) ( cot 33°)

Answer

(1.540)

60) ( cot 140°)

61) ( sec 310° )

Answer

(1.556)

Extensions

For the exercises 62-69, use identities to evaluate the expression.

62) If ( an (t)≈2.7,) and ( sin (t)≈0.94,) find ( cos (t)).

63) If ( an (t)≈1.3,) and ( cos (t)≈0.61), find ( sin (t)).

Answer

( sin (t)≈0.79 )

64) If ( csc (t)≈3.2,) and ( csc (t)≈3.2,) and ( cos (t)≈0.95,) find ( an (t)).

65) If ( cot (t)≈0.58,) and ( cos (t)≈0.5,) find ( csc (t)).

Answer

( csc (t)≈1.16)

66) Determine whether the function (f(x)=2 sin x cos x) is even, odd, or neither.

67) Determine whether the function (f(x)=3 sin ^2 x cos x + sec x) is even, odd, or neither.

Answer

even

68) Determine whether the function (f(x)= sin x −2 cos ^2 x ) is even, odd, or neither.

69) Determine whether the function (f(x)= csc ^2 x+ sec x) is even, odd, or neither.

Answer

even

For the exercises 70-71, use identities to simplify the expression.

70) ( csc t an t)

71) ( dfrac{sec t}{ csc t})

Answer

( dfrac{ sin t}{ cos t}= an t)

Real-World Applications

72) The amount of sunlight in a certain city can be modeled by the function (h=15 cos left(dfrac{1}{600}d ight),) where (h) represents the hours of sunlight, and (d) is the day of the year. Use the equation to find how many hours of sunlight there are on February 10, the (42^{nd}) day of the year. State the period of the function.

73) The amount of sunlight in a certain city can be modeled by the function (h=16 cos left(dfrac{1}{500}d ight)), where (h) represents the hours of sunlight, and (d) is the day of the year. Use the equation to find how many hours of sunlight there are on September 24, the (267^{th}) day of the year. State the period of the function.

Answer

(13.77) hours, period: (1000π)

74) The equation (P=20 sin (2πt)+100) models the blood pressure, (P), where (t) represents time in seconds.

  1. Find the blood pressure after (15) seconds.
  2. What are the maximum and minimum blood pressures?

75) The height of a piston, (h), in inches, can be modeled by the equation (y=2 cos x+6,) where (x) represents the crank angle. Find the height of the piston when the crank angle is (55°).

Answer

(7.73) inches

76) The height of a piston, (h),in inches, can be modeled by the equation (y=2 cos x+5,) where (x) represents the crank angle. Find the height of the piston when the crank angle is (55°).

5.4: Right Triangle Trigonometry

Verbal

1) For the given right triangle, label the adjacent side, opposite side, and hypotenuse for the indicated angle.

Answer

2) When a right triangle with a hypotenuse of (1) is placed in the unit circle, which sides of the triangle correspond to the (x)- and (y)-coordinates?

3) The tangent of an angle compares which sides of the right triangle?

Answer

The tangent of an angle is the ratio of the opposite side to the adjacent side.

4) What is the relationship between the two acute angles in a right triangle?

5) Explain the cofunction identity.

Answer

For example, the sine of an angle is equal to the cosine of its complement; the cosine of an angle is equal to the sine of its complement.

Algebraic

For the exercises 6-9, use cofunctions of complementary angles.

6) ( cos (34°)= sin (\_\_°))

7) ( cos (dfrac{π}{3})= sin (\_\_\_) )

Answer

(dfrac{π}{6})

8) ( csc (21°) = sec (\_\_\_°))

9) ( an (dfrac{π}{4})= cot (\_\_))

Answer

(dfrac{π}{4})

For the exercises 10-16, find the lengths of the missing sides if side (a) is opposite angle (A), side (b) is opposite angle (B), and side (c) is the hypotenuse.

10) ( cos B= dfrac{4}{5},a=10)

11) ( sin B= dfrac{1}{2}, a=20)

Answer

(b= dfrac{20sqrt{3}}{3},c= dfrac{40sqrt{3}}{3})

12) ( an A= dfrac{5}{12},b=6)

13) ( an A=100,b=100)

Answer

(a=10,000,c=10,000.5)

14) (sin B=dfrac{1}{sqrt{3}}, a=2 )

15) (a=5, ∡ A=60^∘)

Answer

(b=dfrac{5sqrt{3}}{3},c=dfrac{10sqrt{3}}{3})

16) (c=12, ∡ A=45^∘)

Graphical

For the exercises 17-22, use Figure below to evaluate each trigonometric function of angle (A).

17) (sin A)

Answer

(dfrac{5sqrt{29}}{29})

18) ( cos A )

19) ( an A )

Answer

(dfrac{5}{2})

20) (csc A )

21) ( sec A )

Answer

(dfrac{sqrt{29}}{2})

22) ( cot A )

For the exercises 23-,28 use Figure below to evaluate each trigonometric function of angle (A).

23) ( sin A)

Answer

(dfrac{5sqrt{41}}{41})

24) ( cos A)

25) ( an A )

Answer

(dfrac{5}{4})

26) ( csc A)

27) ( sec A)

Answer

(dfrac{sqrt{41}}{4})

28) (cot A)

For the exercises 29-31, solve for the unknown sides of the given triangle.

29)

Answer

(c=14, b=7sqrt{3})

30)

31)

Answer

(a=15, b=15 )

Technology

For the exercises 32-41, use a calculator to find the length of each side to four decimal places.

32)

33)

Answer

(b=9.9970, c=12.2041)

34)

35)

Answer

(a=2.0838, b=11.8177)

36)

37) (b=15, ∡B=15^∘)

Answer

(a=55.9808,c=57.9555)

38) (c=200, ∡B=5^∘)

39) (c=50, ∡B=21^∘)

Answer

(a=46.6790,b=17.9184)

40) (a=30, ∡A=27^∘)

41) (b=3.5, ∡A=78^∘)

Answer

(a=16.4662,c=16.8341)

Extensions

42) Find (x).

43) Find (x).

Answer

(188.3159)

44) Find (x).

45) Find (x).

Answer

(200.6737)

46) A radio tower is located (400) feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is (36°), and that the angle of depression to the bottom of the tower is (23°). How tall is the tower?

47) A radio tower is located (325) feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is (43°), and that the angle of depression to the bottom of the tower is (31°). How tall is the tower?

Answer

(498.3471) ft

48) A (200)-foot tall monument is located in the distance. From a window in a building, a person determines that the angle of elevation to the top of the monument is (15°), and that the angle of depression to the bottom of the tower is (2°). How far is the person from the monument?

49) A (400)-foot tall monument is located in the distance. From a window in a building, a person determines that the angle of elevation to the top of the monument is (18°), and that the angle of depression to the bottom of the monument is (3°). How far is the person from the monument?

Answer

(1060.09) ft

50) There is an antenna on the top of a building. From a location (300) feet from the base of the building, the angle of elevation to the top of the building is measured to be (40°). From the same location, the angle of elevation to the top of the antenna is measured to be (43°). Find the height of the antenna.

51) There is lightning rod on the top of a building. From a location (500) feet from the base of the building, the angle of elevation to the top of the building is measured to be (36°). From the same location, the angle of elevation to the top of the lightning rod is measured to be (38°). Find the height of the lightning rod.

Answer

(27.372) ft

Real-World Applications

52) A (33)-ft ladder leans against a building so that the angle between the ground and the ladder is (80°). How high does the ladder reach up the side of the building?

53) A (23)-ft ladder leans against a building so that the angle between the ground and the ladder is (80°). How high does the ladder reach up the side of the building?

Answer

(22.6506) ft

54) The angle of elevation to the top of a building in New York is found to be (9) degrees from the ground at a distance of (1) mile from the base of the building. Using this information, find the height of the building.

55) The angle of elevation to the top of a building in Seattle is found to be (2) degrees from the ground at a distance of (2) miles from the base of the building. Using this information, find the height of the building.

Answer

(368.7633) ft

56) Assuming that a (370)-foot tall giant redwood grows vertically, if I walk a certain distance from the tree and measure the angle of elevation to the top of the tree to be (60°), how far from the base of the tree am I?


5.E: Trigonometric Functions (Exercises) - Mathematics

IDENTITIES, EQUATIONS, AND INEQUALITIES

There are a few trigonometric identities you must know for the Mathematics Level 2 Subject Test.

Reciprocal Identities recognize the definitional relationships:

Cofunction Identities were discussed earlier. Using radian measure:

Pythagorean Identities

Double Angle Formulas

1. Given cos and , find

Since sin 2 = 2(sin )(cos ), you need to determine the value of sin . From the figure below, you can see that sin . Therefore, sin .

2. If cos 23° = z, find the value of cos 46° in terms of z.

Since 46 = 2(23), a double angle formula can be used: cos 2A = 2 cos 2 A – 1. Substituting 23° for A, cos 46° = cos 2(23°) = 2 cos 2 23° – 1 = 2(cos 23°) 2 – 1 = 2z 2 – 1.

3. If sin x = A, find cos 2x in terms of A.

Using the identity cos 2x = 1 – sin 2 x, you get cos 2x = 1 – A 2 .

You may be expected to solve trigonometric equations on the Math Level 2 Subject Test by using your graphing calculator and getting answers that are decimal approximations. To solve any equation, enter each side of the equation into a function (Yn), graph both functions, and find the point(s) of intersection on the indicated domain by choosing an appropriate window.

4. Solve 2 sin x + cos 2x = 2 sin 2 x – 1 for 0 x 2.

Enter 2 sin x + cos 2x into Y1 and 2 sin 2 x – 1 into Y2. Set Xmin = 0, Xmax = 2, Ymin = –4, and Ymax = 4. Solutions (x-coordinates of intersection points) are 1.57, 3.67, and 5.76.

5. Find values of x on the interval [0,] for which cos x 2.62.

1.&emspIf sin and cos , find the value of sin 2x.

&emsp&emsp(A) –

&emsp&emsp(B) –

&emsp&emsp(C)

&emsp&emsp(D)

&emsp&emsp(E)

3.&emspIf cos , find cos 2x.

4.&emspIf sin 37° = z, express sin 74° in terms of z.

&emsp&emsp(A)

&emsp&emsp(E)

5.&emspIf sin x = –0.6427, what is csc x?

6.&emspFor what value(s) of x, 0 0.52

7.&emspWhat is the range of the function f(x) = 5 – 6sin (x + 1)?

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5.E: Trigonometric Functions (Exercises) - Mathematics

If the graph of any trig function f(x) is reflected about the line y = x, the graph of the inverse (relation) of that trig function is the result. Since all trig functions are periodic, graphs of their inverses are not graphs of functions. The domain of a trig function needs to be limited to one period so that range values are achieved exactly once. The inverse of the restricted sine function is sin –1 the inverse of the restricted cosine function is cos –1 , and so forth.

The inverse trig functions are used to represent angles with known trig values. If you know that the tangent of an angle is , but you do not know the degree measure or radian measure of the angle, tan –1 is an expression that represents the angle between whose tangent is .

You can use your graphing calculator to find the degree or radian measure of an inverse trig value.

1. Evaluate the radian measure of tan –1 .

Enter 2nd tan with your calculator in radian mode to get 0.73 radian.

2. Evaluate the degree measure of sin –1 0.8759.

Enter 2nd sin (.8759) with your calculator in degree mode to get 61.15°.

3. Evaluate the degree measure of sec –1 3.4735.

First define x = sec –1 3.4735. If sec x = 3.4735, then cos . Therefore, enter 2nd cos with your calculator in degree mode to get 73.27°.

If “trig” is any trigonometric function, trig(trig –1 x) = x. However, because of the range restriction on inverse trig functions, trig –1 (trig x) need not equal x.

4. Evaluate cos (cos –1 0.72).

5. Evaluate sin –1 (sin 265°).

Enter 2nd sin –1 (sin(265)) with your calculator in degree mode to get –85°. This is because –85° is in the required range [–90°,90°], and –85° has the same reference angle as 265°.

6. Evaluate sin .

Let . Then cos and x is in the first quadrant. See the figure below.

Use the Pythagorean identity sin 2 x + cos 2 x = 1 and the fact that x is in the first quadrant to get sin .

1.&emspFind the number of degrees in .


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Academic dishonesty of any form is unacceptable, and will be reported to the Center for Student Conduct.


Q: A pie comes out of the oven at 325°F and is placed to cool in a 70°F kitchen. The temperature of the.

A: The method of substitution can be used to obtain the time required to cool down. So, substitute T=11.

Q: To determine drug dosages, doctors estimate a person’s body surface area (BSA) (in meters squared) u.

A: Solution:Given BSA=hm60To calculate the rate of change of BSA with respect to mass for a person.

A: a+b=c Where a=addend , b= addend , c= sum or total

Q: QUESTION 12 Evaluate the integral by changing the order of integration in an appropriate way. x sin .

A: Click to see the answer

Q: 4. Find the average rate of change of f(x) = 3x - 5 from x = 0 to x = 4.

A: Click to see the answer

Q: A particle is thrown vertically upwards in the air. The distance it covers in time t is given by s(t.

A: We will be using the fact that rate of change in distance is called velocity. v(t)=dsdt

A: The domain of a function is set of values for which the function is defined. For a logarithmic funct.

Q: 50. Is the function given by f(x) = 3x - 2 continuous at x = 5? Why or why not?

A: To Check: Whether the function fx=3x-2 is continuous at x=5. Concept used: If the function fx is con.


Preface to the Instructor xv

Preface to the Student xxiii

Chapter 0 The Real Numbers 1

Construction of the Real Line 2

Is Every Real Number Rational? 3

0.2 Algebra of the Real Numbers 6

Commutativity and Associativity 6

The Order of Algebraic Operations 7

The Distributive Property 8

Additive Inverses and Subtraction 9

Multiplicative Inverses and the Algebra of Fractions 10

Exercises, Problems, and Worked-out Solutions 15

0.3 Inequalities, Intervals, and Absolute Value 20

Positive and Negative Numbers 20

Exercises, Problems, and Worked-out Solutions 29

Chapter Summary and Chapter Review Questions 35

Chapter 1 Functions and Their Graphs 37

Definition and Examples 38

The Domain of a Function 41

The Range of a Function 42

Exercises, Problems, and Worked-out Solutions 45

1.2 The Coordinate Plane and Graphs 50

The Graph of a Function 52

Determining the Domain and Range from a Graph 54

Which Sets are Graphs of Functions? 56

Exercises, Problems, and Worked-out Solutions 56

1.3 Function Transformations and Graphs 63

Vertical Transformations: Shifting, Stretching, and Flipping 63

Horizontal Transformations: Shifting, Stretching, Flipping 66

Combinations of Vertical Function Transformations 68

Exercises, Problems, and Worked-out Solutions 73

1.4 Composition of Functions 81

Combining Two Functions 81

Definition of Composition 82

Composing More than Two Functions 85

Function Transformations as Compositions 86

Exercises, Problems, and Worked-out Solutions 88

The Definition of an Inverse Function 95

The Domain and Range of an Inverse Function 97

The Composition of a Function and Its Inverse 98

Comments About Notation 99

Exercises, Problems, and Worked-out Solutions 101

1.6 A Graphical Approach to Inverse Functions 106

The Graph of an Inverse Function 106

Graphical Interpretation of One-to-One 107

Increasing and Decreasing Functions 108

Inverse Functions via Tables 110

Exercises, Problems, and Worked-out Solutions 111

Chapter Summary and Chapter Review Questions 115

Chapter 2 Linear, Quadratic, Polynomial, and Rational Functions 119


Calculus Lab - SDSU edition 1st edition

Your students have access to an online version of the textbook that might contain additional interactive features.

Access is contingent on use of this textbook in the instructor's classroom.

  • Chapter 0: Prerequisite Material
    • 0.P: Practice Problems (16)
    • 0.E: Exercises (4)
    • 1.P: Practice Problems (12)
    • 1.E: Exercises (10)
    • 2.P: Practice Problems (4)
    • 2.E: Exercises (6)
    • 3.P: Practice Problems (7)
    • 3.E: Exercises (20)
    • 4.P: Practice Problems (4)
    • 4.E: Exercises (7)
    • 5.P: Practice Problems (4)
    • 5.E: Exercises (36)
    • 6.P: Practice Problems (1)
    • 6.E: Exercises (4)
    • 7.P: Practice Problems (1)
    • 7.E: Exercises (8)
    • 8.P: Practice Problems (2)
    • 8.E: Exercises (7)
    • 9.P: Practice Problems (2)
    • 9.E: Exercises (4)
    • 10.P: Practice Problems (5)
    • 10.E: Exercises (12)
    • 11.P: Practice Problems (1)
    • 11.E: Exercises (6)
    • 12.P: Practice Problems (6)
    • 12.E: Exercises (10)

    The SDSU Calculus Lab manual is designed to help supplement any first semester calculus course. The goal of this lab is to help your students improve their algebra and trigonometry skills as needed to be better prepared for first semester calculus. Encourage your students to take this lab manual seriously in order to achieve the highest success in calculus. This means reading all lab materials, watching the videos, completing the practice problems, and completing other assigned problems, reminding them that the most important way to become better at mathematics is to practice.


    Chapter 1

    The degree is 6, the leading term is − x 6 , − x 6 , and the leading coefficient is −1. −1.

    3 x 4 −10 x 3 −8 x 2 + 21 x + 14 3 x 4 −10 x 3 −8 x 2 + 21 x + 14

    6 x 2 + 21 x y −29 x −7 y + 9 6 x 2 + 21 x y −29 x −7 y + 9

    1.5 Factoring Polynomials

    ( 6 a + b ) ( 36 a 2 −6 a b + b 2 ) ( 6 a + b ) ( 36 a 2 −6 a b + b 2 )

    ( 10 x − 1 ) ( 100 x 2 + 10 x + 1 ) ( 10 x − 1 ) ( 100 x 2 + 10 x + 1 )

    1.6 Rational Expressions

    ( x + 5 ) ( x + 6 ) ( x + 2 ) ( x + 4 ) ( x + 5 ) ( x + 6 ) ( x + 2 ) ( x + 4 )

    1.1 Section Exercises

    irrational number. The square root of two does not terminate, and it does not repeat a pattern. It cannot be written as a quotient of two integers, so it is irrational.

    The Associative Properties state that the sum or product of multiple numbers can be grouped differently without affecting the result. This is because the same operation is performed (either addition or subtraction), so the terms can be re-ordered.

    inverse property of addition

    1.2 Section Exercises

    No, the two expressions are not the same. An exponent tells how many times you multiply the base. So 2 3 2 3 is the same as 2 × 2 × 2 , 2 × 2 × 2 , which is 8. 3 2 3 2 is the same as 3 × 3 , 3 × 3 , which is 9.

    It is a method of writing very small and very large numbers.

    1.3 Section Exercises

    When there is no index, it is assumed to be 2 or the square root. The expression would only be equal to the radicand if the index were 1.

    The principal square root is the nonnegative root of the number.

    1.4 Section Exercises

    The statement is true. In standard form, the polynomial with the highest value exponent is placed first and is the leading term. The degree of a polynomial is the value of the highest exponent, which in standard form is also the exponent of the leading term.

    Use the distributive property, multiply, combine like terms, and simplify.

    11 b 4 −9 b 3 + 12 b 2 −7 b + 8 11 b 4 −9 b 3 + 12 b 2 −7 b + 8

    16 t 4 + 4 t 3 −32 t 2 − t + 7 16 t 4 + 4 t 3 −32 t 2 − t + 7

    32 t 3 − 100 t 2 + 40 t + 38 32 t 3 − 100 t 2 + 40 t + 38

    a 4 + 4 a 3 c −16 a c 3 −16 c 4 a 4 + 4 a 3 c −16 a c 3 −16 c 4

    1.5 Section Exercises

    The terms of a polynomial do not have to have a common factor for the entire polynomial to be factorable. For example, 4 x 2 4 x 2 and −9 y 2 −9 y 2 don’t have a common factor, but the whole polynomial is still factorable: 4 x 2 −9 y 2 = ( 2 x + 3 y ) ( 2 x −3 y ) . 4 x 2 −9 y 2 = ( 2 x + 3 y ) ( 2 x −3 y ) .

    ( 5 a + 7 ) ( 25 a 2 − 35 a + 49 ) ( 5 a + 7 ) ( 25 a 2 − 35 a + 49 )

    ( 4 x − 5 ) ( 16 x 2 + 20 x + 25 ) ( 4 x − 5 ) ( 16 x 2 + 20 x + 25 )

    ( 5 r + 12 s ) ( 25 r 2 − 60 r s + 144 s 2 ) ( 5 r + 12 s ) ( 25 r 2 − 60 r s + 144 s 2 )

    ( 4 z 2 + 49 a 2 ) ( 2 z + 7 a ) ( 2 z − 7 a ) ( 4 z 2 + 49 a 2 ) ( 2 z + 7 a ) ( 2 z − 7 a )

    1 ( 4 x + 9 ) ( 4 x −9 ) ( 2 x + 3 ) 1 ( 4 x + 9 ) ( 4 x −9 ) ( 2 x + 3 )

    1.6 Section Exercises

    You can factor the numerator and denominator to see if any of the terms can cancel one another out.

    True. Multiplication and division do not require finding the LCD because the denominators can be combined through those operations, whereas addition and subtraction require like terms.

    3 c 2 + 3 c − 2 2 c 2 + 5 c + 2 3 c 2 + 3 c − 2 2 c 2 + 5 c + 2

    Review Exercises

    ( 4 q − 3 p ) ( 16 q 2 + 12 p q + 9 p 2 ) ( 4 q − 3 p ) ( 16 q 2 + 12 p q + 9 p 2 )

    Practice Test

    ( 3 c − 11 ) ( 9 c 2 + 33 c + 121 ) ( 3 c − 11 ) ( 9 c 2 + 33 c + 121 )

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      Q: sketch the graph of the function. Indicate the transition points and asymptotes

      A: Click to see the answer

      Q: 16. Gasoline in a tank A gasoline tank is in the shape of a right circular cylinder (lying on its si.

      A: Given information: The length of the right circular cylinder tank is 10 ft. The radius of the right .

      Q: State whether f (2) and f (4) are local minima or local maxima, assuming that Figure 14 is the graph.

      A: To identify whether f(2) and f(4) are local maxima or local minima.

      Q: 115. Suppose that f(x) dx = 3. Find f(x) dx if a. f is odd, b. f is even. you

      A: Click to see the answer

      A: There is typo mistake in the question. Consider that the gravel is being dumped from a conveyor belt.

      Q: Tutorial Exercise Use the given transformation to evaluate the given integral, where R is the triang.

      A: To determine the Jacobian of the transformation.

      Q: Find the equation of the tangent line to the graph of the function f (x) = v 4,4 + 12 at the point (.

      Q: 51. The functions y = e* and y = x³e do not have elementary anti- derivatives, but y = (1 + 3x)e" do.

      A: Click to see the answer

      Q: help pls and thanks. find the equation of the line tangent to. f(x)= (2x^2-x)/x at point (1,3)


      With Videos

      Explanatory Answer

      How to compute HCF by Prime Factorisation Method?

      HCF of two numbers is the product of LOWEST power of COMMON primes found in the numbers.

      How to find LCM by Prime Factorisation Method?

      LCM of two numbers is the product of HIGHEST power of ALL primes found in the numbers.

      (i)LCM and HCF of 2 5 × 5 4 × 7 2 × 13 6 and 2 3 × 5 6 × 7 × 17 3

      Notice that the two numbers are already prime factorized. That makes life really easy.
      Let us compute HCF first.
      The common primes in the two numbers are 2, 5, and 7.
      The lowest power of 2, 5, and 7 are respectively 2 3 , 5 4 , and 7 1
      ∴ HCF of 2 5 × 5 4 × 7 2 × 13 6 and 2 3 × 5 6 × 7 × 17 3 is 2 3 × 5 4 × 7

      Let us compute LCM next.
      The primes found in the two numbers are 2, 5, 7, 13, and 17.
      The highest power of 2, 5, 7, 13, and 17 are 2 5 , 5 6 , 7 2 , 13 6 , and 17 3 respectively.
      ∴ LCM of 2 5 × 5 4 × 7 2 × 13 6 and 2 3 × 5 6 × 7 × 17 3 is 2 5 × 5 6 × 7 2 × 13 6 × 17 3

      (ii) Find the LCM and HCF of a 5 × b 2 × c 2 × d 5 and a 7 × b 3 × e × f 3

      Given: a, b, c, d, e, and f are prime numbers.
      Let us compute HCF first.
      The common primes in the two numbers are a and b.
      The lowest power of a and b in the two numbers are a 5 and b 2 respectively.
      ∴ HCF of a 5 × b 2 × c 2 × d 5 and a 7 × b 3 × e × f 3 is a 5 × b 2

      Let us compute LCM next.
      The primes found in the two numbers are a, b, c, d, e, and f.
      The highest power of a, b, c, d, e, and f are a 7 , b 3 , c 2 , d 5 , e 1 , and f 3 respectively.
      ∴ LCM of a 5 × b 2 × c 2 × d 5 and a 7 × b 3 × e × f 3 is a 7 × b 3 × c 2 × d 5 × e × f 3


      Watch the video: ΜΑΘΗΜΑΤΙΚΑ Β ΛΥΚΕΙΟΥ: Βασικές Τριγωνομετρικές Ταυτότητες Ασκήσεις part IIΙ (December 2021).