text: # Trigonometry
  
  1. ### If the radius of a pizza is 20 cm, what is the central angle in degrees that gives one person 30 cm of crust?
  
  Question 5 options: a) 120° b) 35° c) 43° d) 86° I say option a, but my partner disagrees
  
  6. ### Hello!
  
  Explain why the primary trigonometric ratios depend only on the given angle and not the size of legs and hypotenuse of a right triangle? I am not 100% sure but it is because the angles have the same value for all of the ratios? Is it also because the
  
  8. ### If the central angle is 4pi/3 radians, what should the radius of a circle be to make the arc length 1 m?
  
  a) 0.424 m b) 0.238 m c) 2.356 m d) 4.188 m I think it is c)... is that correct?
  
  10. ### Find the value of x. Round to the nearest tenth. The diagram is not drawn to scale.
  
  The adjacent is 12cm wide and the angle of the hypotenuse is at 35 degrees. The opposite is x. I'm posting this here because I want to know what I did wrong regarding my
  
  12. ### Use a graph of the function to approximate the solution of the equation on the interval
  
  [−2π, 2π]. (List the solutions in increasing order from left to right on the x-axis. Round your answers to three decimal places.) cot x = −1
  
  16. ### The graph of y=tan x has vertical asymptotes at certain values of x because the tangent ratio is _____ at those values.
  
  a) vertical b) undefined c) zero d) intermediate I believe it's a), but I am not too sure.
  
  19. ### How do you verify the equation is an identity?
  
  Tan^2x-tan^2y=sec^2x-sec^2y and, how do you factor and simplify, cscx(sin^2x+cos^2xtanx)/sinx+cosx
  
  21. ### A bridge across a valley is 150 m in
  
  length. The valley walls make angles of 60° and 54° with the bridge that spans it, as shown. How deep is the valley, to the nearest metre?
  
  22. ### Prove identity.
  
  Sec(pi-x)=-sec x
  
  cos((9π/2)-θ
  
  24. ### Find all solutions of the equation in the interval [0, 2π). (Enter your answers as a comma-separated list.)
  
  sin(x + π) − sin x + sqrt(3)=0
  
  30. ### 2cos^2(x) = 13sinx - 5
  
  How do i solve for x? I think i have to use the sin^2x + cos^2x = 1, but I'm not sure how to use that. I thought about maybe doing 2(1-sin^2x) = 13sinx - 5 but then i get stuck. Any suggestions?
  
  31. ### Surveying
  
  A surveyor wishes to find the distance across a swamp. The bearing from A to B (Segment AB is opposite side of triangle) is N 32° W. The surveyor walks 50 meters from A to C, and at the point C the bearing to B is N 68° W. (Segment AC is
  
  35. ### Stumped on this question - please explain!
  
  If sin A = 0.35, cos A = 0.94, sin B = 0.58, and cos B = 0.81, what is sin(A + B)? Am I just plugging the values into sin(A + B) or do I have to do more than that? Thank you.
  
  37. ### Find all solutions between 0 and 2pi. Round to two decimal places. In radians.
  
  Find all solutions between 0 and 2 pi. Round to two decimal places for the final solutions. The answers should be in radian mode. If you can use exact values use them. At least
  
  39. ### prove that [(sin 2 t / sin t )] - [( cos 2t ) / cos t ] = sec t
  
  and sin (2t - t ) = sin t
  
  41. ### If someone could tell me if this is correct, it would really help me out.
  
  Problem: A statue 20 feet high stands on top of a base. From a point in front of the statue, the angle of elevation to the top of the statue is 48 degrees, and the angle of elevation
  
  42. ### Find all solutions of 4 (cos (x)**2)-1=0 in the interval (6pi, 8pi).
  
  (Leave your answers in exact form and enter them as a comma-separated list.)
  
  45. ### Given the trigonometric function y = tan x, find the x-coordinate of the point (3pi/4, ?)
  
  a) -1 b) root 2 c) c d) 1/root 2
  
  46. ### a bicycle with 24inch diameter wheels is traveling at 15mi/hr.
  
  a) Find the angular speed of the wheels in rad/min. b)How many revolutions per minute do the wheels make?
  
  50. ### Solve the equation. (Find all solutions of the equation in the interval [0, 2π). Enter your answers as a comma-separated list.)
  
  4 tan(2x) − 4 cot(x) = 0
  
  53. ### Verify the identity algebraically.
  
  TAN X + COT Y/TAN X COT Y= TAN Y + COT X
  
  54. ### over which of the following domains is f(x) = csc(0.1x + 1.2) defined at all points and invertible?
  
  x = [0,10] x = [10,20] x = [20,30] x = [30,40] Can someone please explain how to solve this to me and show me all the steps rather than simply giving me the
  
  57. ### Determine the number of triangles ABC possible with the given parts.
  
  b=60 a=82 B=100
  
  58. ### Hi! The question is really confusing to me. I can't sketch/visualize the problem. :(
  
  From a given position an observer notes that the angle of elevation of a rock is 47 degrees. After walking 1000 feet towards the rock, up a slope of 32 degrees, he finds
  
  64. ### Find the diameter of a pulley which is driven at 360 rpm by a belt moving at 40 ft/s.
  
  Then in 1 s the pulley turns through an angle theta measuring 12 pi radians and a point on the rim travels a distance s= 40 ft.
  
  66. ### Find all degree solutions. (Enter your answers as a comma-separated list. Let k be any integer.)
  
  2 cos2 6θ + 3 cos 6θ + 1 = 0
  
  67. ### The expression log(x^n/ radical y)is equivalent to
  
  1. n log x - 1/2 log y 2. n log x- 2 log y 3. log (nx) - log (1/2y) 4. log (nx) - log (2y)
  
  68. ### A plane is observed approaching your home and you assume its speed to be 550 miles per hour. The angle of elevation of the plane is 16 degrees and one minute later is 57 degrees. Approximate the altitude of the plane.
  
  Draw yourself a figure with two
  
  71. ### A sine function has an amplitude of 4/7, period of 2pi, horizontal shift of -3pi, and vertical shift of 1.
  
  What is the y-value of the positive function at x= pi/2?
  
  72. ### which of the following is a solution for the equation 1/485 tan^2x=0
  
  a) no solution b) 485 pi c) pi d) pi/2 I'm getting conflicting answers when I try to solve it.
  
  7. What is 1/cot(x) in terms of sine? This is what I've got so far: sin = 1/csc tan = 1/cot 1+cot2(x) = csc2(x) 1/1+cot2(x) = 1/csc2(x)
  
  76. ### If A + B + C = 180°, Prove that
  
  Cos²A + Cos²B + Cos²C = 1-2cosAcosBcosC
  
  78. ### State the quadrant in which theta lies.
  
  sin(theta) > 0 and cos(theta) > 0 How can I determine this? Please explain.
  
  79. ### The populations, P, of six towns with time t in years are given by:
  
  I) P=1000(1.08)^t II) P= 600(1.12)^t III) P = 2500(0.9)^t IV) P=1200(1.185)^t V) P=800(0.78)^t VI) P=2000(0.99)^t a. Which towns are growing in size? Which are shrinking? b. Which town is
  
  80. ### If the terminal side of angle theta passes through point (-3, -4), what is the valuse of sec theta?
  
  1) 5/3 2) -5/3 3)5/4 4)-5/4
  
  86. ### Prove that
  
  cos(A+B) + sin(A-B) = 2sin(45°+A)cos(45°+B)
  
  87. ### This is a logs question
  
  If u=x/y^2, which expression is equivalent to log u? 1) log x + 2 log y 2) 2(log x- log y) 3) 2(log x + log y) 4) log x- 2 log y
  
  91. ### 4. Find the exact value for sin(x+y) if sinx=-4/5 and cos y = 15/17. Angles x and y are in the fourth quadrant.
  
  5. Find the exact value for cos 165degrees using the half-angle identity. 1. Solve: 2 cos^2x - 3 cosx + 1 = 0 for 0 less than or equal to x <2pi.
  
  92. ### A Ferris Wheel rotates 3 times each minute. The passengers sit in seats that are 25 feet from the center of the wheel.
  
  A. What is the angular velocity of the wheel in degrees per minute and radians per minute? B. What is the linear velocity of the passenger
  
  93. ### Angle x is a third quadrant angle such that cos x = -(2/3).
  
  What is the exact value of cos (x/2) ? Solve in simplest radical form.
  
  94. ### How do I find all solutions to:
  
  tanx+3cotx=4? The answer choices are: a.) x=arctan3+nPi b.) x=arccot3+nPi
  
  95. ### A 19-ft ladder leans against a building so that the angle between the ground and the ladder is 80∘.
  
  How high does the ladder reach on the building? I did 19sin(80) to get me 18.76847847 for how high the ladder reach on the building... but when i put the
  
  96. ### 1. 4sin^2x=3tan^2x-1
  
  2. 8cos^2x-4cos^4x=3 3. 3secx-cosx=2
  
  99. ### Having trouble with true/false questions in Trigonometry. They read as follows -
  
  True or False: For a trigonometric function, y=f(x), then x=F^-1(y). Explain your answer. True or False: For a one to one function, y=f(x), then x=f^-1(y). Explain your answer.
explanation: The document contains multiple trigonometry questions and some answers provided by the user. Each question is numbered and covers various topics such as angles, trigonometric identities, and solving trigonometric equations. Some questions are followed by the user's attempt at an answer, while others are left unanswered.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Trigonometry
  
  1. ### If the radius of a pizza is 20 cm, what is the central angle in degrees that gives one person 30 cm of crust?
  
  Question 5 options: a) 120° b) 35° c) 43° d) 86° I say option a, but my partner disagrees
  
  6. ### Hello!
  
  Explain why the primary trigonometric ratios depend only on the given angle and not the size of legs and hypotenuse of a right triangle? I am not 100% sure but it is because the angles have the same value for all of the ratios? Is it also because the
  
  8. ### If the central angle is 4pi/3 radians, what should the radius of a circle be to make the arc length 1 m?
  
  a) 0.424 m b) 0.238 m c) 2.356 m d) 4.188 m I think it is c)... is that correct?
  
  10. ### Find the value of x. Round to the nearest tenth. The diagram is not drawn to scale.
  
  The adjacent is 12cm wide and the angle of the hypotenuse is at 35 degrees. The opposite is x. I'm posting this here because I want to know what I did wrong regarding my
  
  12. ### Use a graph of the function to approximate the solution of the equation on the interval
  
  [−2π, 2π]. (List the solutions in increasing order from left to right on the x-axis. Round your answers to three decimal places.) cot x = −1
  
  16. ### The graph of y=tan x has vertical asymptotes at certain values of x because the tangent ratio is _____ at those values.
  
  a) vertical b) undefined c) zero d) intermediate I believe it's a), but I am not too sure.
  
  19. ### How do you verify the equation is an identity?
  
  Tan^2x-tan^2y=sec^2x-sec^2y and, how do you factor and simplify, cscx(sin^2x+cos^2xtanx)/sinx+cosx
  
  21. ### A bridge across a valley is 150 m in
  
  length. The valley walls make angles of 60° and 54° with the bridge that spans it, as shown. How deep is the valley, to the nearest metre?
  
  22. ### Prove identity.
  
  Sec(pi-x)=-sec x
  
  cos((9π/2)-θ
  
  24. ### Find all solutions of the equation in the interval [0, 2π). (Enter your answers as a comma-separated list.)
  
  sin(x + π) − sin x + sqrt(3)=0
  
  30. ### 2cos^2(x) = 13sinx - 5
  
  How do i solve for x? I think i have to use the sin^2x + cos^2x = 1, but I'm not sure how to use that. I thought about maybe doing 2(1-sin^2x) = 13sinx - 5 but then i get stuck. Any suggestions?
  
  31. ### Surveying
  
  A surveyor wishes to find the distance across a swamp. The bearing from A to B (Segment AB is opposite side of triangle) is N 32° W. The surveyor walks 50 meters from A to C, and at the point C the bearing to B is N 68° W. (Segment AC is
  
  35. ### Stumped on this question - please explain!
  
  If sin A = 0.35, cos A = 0.94, sin B = 0.58, and cos B = 0.81, what is sin(A + B)? Am I just plugging the values into sin(A + B) or do I have to do more than that? Thank you.
  
  37. ### Find all solutions between 0 and 2pi. Round to two decimal places. In radians.
  
  Find all solutions between 0 and 2 pi. Round to two decimal places for the final solutions. The answers should be in radian mode. If you can use exact values use them. At least
  
  39. ### prove that [(sin 2 t / sin t )] - [( cos 2t ) / cos t ] = sec t
  
  and sin (2t - t ) = sin t
  
  41. ### If someone could tell me if this is correct, it would really help me out.
  
  Problem: A statue 20 feet high stands on top of a base. From a point in front of the statue, the angle of elevation to the top of the statue is 48 degrees, and the angle of elevation
  
  42. ### Find all solutions of 4 (cos (x)**2)-1=0 in the interval (6pi, 8pi).
  
  (Leave your answers in exact form and enter them as a comma-separated list.)
  
  45. ### Given the trigonometric function y = tan x, find the x-coordinate of the point (3pi/4, ?)
  
  a) -1 b) root 2 c) c d) 1/root 2
  
  46. ### a bicycle with 24inch diameter wheels is traveling at 15mi/hr.
  
  a) Find the angular speed of the wheels in rad/min. b)How many revolutions per minute do the wheels make?
  
  50. ### Solve the equation. (Find all solutions of the equation in the interval [0, 2π). Enter your answers as a comma-separated list.)
  
  4 tan(2x) − 4 cot(x) = 0
  
  53. ### Verify the identity algebraically.
  
  TAN X + COT Y/TAN X COT Y= TAN Y + COT X
  
  54. ### over which of the following domains is f(x) = csc(0.1x + 1.2) defined at all points and invertible?
  
  x = [0,10] x = [10,20] x = [20,30] x = [30,40] Can someone please explain how to solve this to me and show me all the steps rather than simply giving me the
  
  57. ### Determine the number of triangles ABC possible with the given parts.
  
  b=60 a=82 B=100
  
  58. ### Hi! The question is really confusing to me. I can't sketch/visualize the problem. :(
  
  From a given position an observer notes that the angle of elevation of a rock is 47 degrees. After walking 1000 feet towards the rock, up a slope of 32 degrees, he finds
  
  64. ### Find the diameter of a pulley which is driven at 360 rpm by a belt moving at 40 ft/s.
  
  Then in 1 s the pulley turns through an angle theta measuring 12 pi radians and a point on the rim travels a distance s= 40 ft.
  
  66. ### Find all degree solutions. (Enter your answers as a comma-separated list. Let k be any integer.)
  
  2 cos2 6θ + 3 cos 6θ + 1 = 0
  
  67. ### The expression log(x^n/ radical y)is equivalent to
  
  1. n log x - 1/2 log y 2. n log x- 2 log y 3. log (nx) - log (1/2y) 4. log (nx) - log (2y)
  
  68. ### A plane is observed approaching your home and you assume its speed to be 550 miles per hour. The angle of elevation of the plane is 16 degrees and one minute later is 57 degrees. Approximate the altitude of the plane.
  
  Draw yourself a figure with two
  
  71. ### A sine function has an amplitude of 4/7, period of 2pi, horizontal shift of -3pi, and vertical shift of 1.
  
  What is the y-value of the positive function at x= pi/2?
  
  72. ### which of the following is a solution for the equation 1/485 tan^2x=0
  
  a) no solution b) 485 pi c) pi d) pi/2 I'm getting conflicting answers when I try to solve it.
  
  7. What is 1/cot(x) in terms of sine? This is what I've got so far: sin = 1/csc tan = 1/cot 1+cot2(x) = csc2(x) 1/1+cot2(x) = 1/csc2(x)
  
  76. ### If A + B + C = 180°, Prove that
  
  Cos²A + Cos²B + Cos²C = 1-2cosAcosBcosC
  
  78. ### State the quadrant in which theta lies.
  
  sin(theta) > 0 and cos(theta) > 0 How can I determine this? Please explain.
  
  79. ### The populations, P, of six towns with time t in years are given by:
  
  I) P=1000(1.08)^t II) P= 600(1.12)^t III) P = 2500(0.9)^t IV) P=1200(1.185)^t V) P=800(0.78)^t VI) P=2000(0.99)^t a. Which towns are growing in size? Which are shrinking? b. Which town is
  
  80. ### If the terminal side of angle theta passes through point (-3, -4), what is the valuse of sec theta?
  
  1) 5/3 2) -5/3 3)5/4 4)-5/4
  
  86. ### Prove that
  
  cos(A+B) + sin(A-B) = 2sin(45°+A)cos(45°+B)
  
  87. ### This is a logs question
  
  If u=x/y^2, which expression is equivalent to log u? 1) log x + 2 log y 2) 2(log x- log y) 3) 2(log x + log y) 4) log x- 2 log y
  
  91. ### 4. Find the exact value for sin(x+y) if sinx=-4/5 and cos y = 15/17. Angles x and y are in the fourth quadrant.
  
  5. Find the exact value for cos 165degrees using the half-angle identity. 1. Solve: 2 cos^2x - 3 cosx + 1 = 0 for 0 less than or equal to x <2pi.
  
  92. ### A Ferris Wheel rotates 3 times each minute. The passengers sit in seats that are 25 feet from the center of the wheel.
  
  A. What is the angular velocity of the wheel in degrees per minute and radians per minute? B. What is the linear velocity of the passenger
  
  93. ### Angle x is a third quadrant angle such that cos x = -(2/3).
  
  What is the exact value of cos (x/2) ? Solve in simplest radical form.
  
  94. ### How do I find all solutions to:
  
  tanx+3cotx=4? The answer choices are: a.) x=arctan3+nPi b.) x=arccot3+nPi
  
  95. ### A 19-ft ladder leans against a building so that the angle between the ground and the ladder is 80∘.
  
  How high does the ladder reach on the building? I did 19sin(80) to get me 18.76847847 for how high the ladder reach on the building... but when i put the
  
  96. ### 1. 4sin^2x=3tan^2x-1
  
  2. 8cos^2x-4cos^4x=3 3. 3secx-cosx=2
  
  99. ### Having trouble with true/false questions in Trigonometry. They read as follows -
  
  True or False: For a trigonometric function, y=f(x), then x=F^-1(y). Explain your answer. True or False: For a one to one function, y=f(x), then x=f^-1(y). Explain your answer.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains multiple trigonometry questions and some answers provided by the user. Each question is numbered and covers various topics such as angles, trigonometric identities, and solving trigonometric equations. Some questions are followed by the user's attempt at an answer, while others are left unanswered.
  
  Label: multiple-question-and-answer