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B.Sc Mathematics Trigonometric Identities and Equations — practice questions — page 3

Explore 73 free Trigonometric Identities and Equations questions in B.Sc Mathematics Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Trigonometric Identities and Equations questions and exercises with answers

Showing questions 61–73 of 73.

How many solutions does 2 sin^2 t - 1 = 0 have on the interval from 0 to 2 pi?Read the answer and worked solution →Solving csc t = -2 on the interval from 0 to 4 pi, in which quadrants do all four solutions lie?Read the answer and worked solution →A calculator solving sin t = 0.8 returns an angle in which quadrants only?Read the answer and worked solution →With sin t = 0.8 and the first angle at about 53.1 degrees, what is the second angle within one turn?Read the answer and worked solution →Solving sec t = -4 gives t about 1.8235 radians. What is the other solution within one turn?Read the answer and worked solution →Solve 2 sin^2 t + sin t = 0 exactly on the interval from 0 to 2 pi.Read the answer and worked solution →Solve cos t cos(2t) + sin t sin(2t) = sqrt(3)/2 on the interval from 0 to 2 pi.Read the answer and worked solution →Solve 3 cos t + 3 = 2 sin^2 t exactly on the interval from 0 to 2 pi.Read the answer and worked solution →Solve cos(2t) = 1/2 on the interval from 0 to 2 pi.Read the answer and worked solution →Solving sin(n t) = c across one turn requires going around the unit circle how many times?Read the answer and worked solution →Compared with y = sin t, the graph of y = sin(2t) is which transformation?Read the answer and worked solution →Safety rules place a ladder's base 1 foot from the wall for every 4 feet of ladder length. What angle does the ladder make with the ground?Read the answer and worked solution →If a ladder's base is a feet from the wall and its length is 4a feet, how high up the wall does it reach?Read the answer and worked solution →

Worked practice questions

How many solutions does 2 sin^2 t - 1 = 0 have on the interval from 0 to 2 pi?

  1. Two solutions
  2. Three solutions
  3. Four solutions
  4. Six solutions
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Correct answer: C. Four solutions

1. Isolate the square: 2 sin^2 t = 1, so sin^2 t = 1/2. 2. The square root property gives sin t = plus or minus 1 / sqrt(2), that is plus or minus sqrt(2)/2. 3. The positive value gives the quadrant I and quadrant II angles pi/4 and 3 pi/4. 4. The negative value gives the quadrant III and quadrant IV angles 5 pi/4 and 7 pi/4. 5. That is four solutions in total, and answering two comes from keeping only the positive root. _Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 9 "Trigonometric Identities and Equations", section 9.5 Solving Trigonometric Equations_

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Solving csc t = -2 on the interval from 0 to 4 pi, in which quadrants do all four solutions lie?

  1. The first and second quadrants
  2. The second and third quadrants
  3. The third and fourth quadrants
  4. The first and fourth quadrants
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Correct answer: C. The third and fourth quadrants

1. Write csc t = -2 as 1 / sin t = -2. 2. Taking reciprocals gives sin t = -1/2. 3. A negative sine puts the angle in quadrant III or quadrant IV. 4. In the first turn the solutions are 7 pi/6 and 11 pi/6. 5. The interval runs to 4 pi, so the second turn adds 19 pi/6 and 23 pi/6. 6. All four solutions therefore lie in the third and fourth quadrants. 7. Quadrants I and II would require a positive sine, which contradicts the negative cosecant. _Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 9 "Trigonometric Identities and Equations", section 9.5 Solving Trigonometric Equations_

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A calculator solving sin t = 0.8 returns an angle in which quadrants only?

  1. The first or the second quadrant
  2. The second or the third quadrant
  3. The third or the fourth quadrant
  4. The first or the fourth quadrant
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Correct answer: D. The first or the fourth quadrant

1. The inverse sine key returns values only within the range of the inverse sine function. 2. That range covers quadrant I for positive inputs and quadrant IV for negative ones. 3. So a calculator never hands back a quadrant II or quadrant III angle for sine. 4. For sin t = 0.8 it returns about 0.9273 radians, and the second angle must be found separately as pi minus that value. _Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 9 "Trigonometric Identities and Equations", section 9.5 Solving Trigonometric Equations_

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