Learn – Algebra/Precalculus – Trigonometry

Analytic trigonometry

Exercises

Problem set

Evaluate the following.

  1. \sin 75^\circ
  2. \sin 105^\circ
  3. \sin \frac{7\pi}{12}
  4. \cos 75^\circ
  5. \cos 105^\circ
  6. \cos \frac{5\pi}{12}
  7. \sin 15^\circ
  8. \cos \frac{\pi}{12}

Problem set

Evaluate the following.

  1. \sin 195^\circ
  2. \cos \left(-255^\circ\right)
  3. \cos \frac{49\pi}{12}
  4. \sin \left(-\frac{49\pi}{12}\right)

Problem set

Simplify each of the following.

  1. \csc\theta\tan \theta
  2. \sec\theta\cot\theta
  3. \cos x+\sin x \cot x
  4. \csc x+\sec x \cot x
  5. \sin x+\cos x \cot x

Problem set

Simplify each of the following.

  1. \csc x+\sec x \tan x
  2. \sec^2 x+\csc^2 x
  3. \frac{1}{1+\tan^2\theta}
  4. \frac{1+\tan^2\theta}{1+\cot^2\theta}
  5. \cos \alpha \sec \alpha - \cos \alpha \cos \alpha
  6. (\sec \theta + 1)(\sec \theta - 1)

Problem set

Justify the following algebraically.

  1. \cos\theta + \sin\theta\tan\theta = \sec\theta
  2. \frac{1}{1-\sin\theta}+\frac{1}{1+\sin\theta} = 2\sec^2\theta
  3. \frac{\cos^2\theta}{1-\sin\theta} - 1 = \sin\theta
  4. \frac{\cos\theta}{1+\sin\theta}+\frac{1+\sin\theta}{\cos\theta} = 2\sec\theta
  5. \csc\theta + \frac{1}{\cot\theta - \csc\theta} = -\cot\theta

Problem set

Justify the following algebraically.

  1. \frac{\csc\theta - \sin\theta}{\cot\theta} = \cos\theta
  2. \frac{\sec\theta - \cos\theta}{\sin\theta} = \tan\theta
  3. \tan(\theta+\phi) = \frac{\tan\theta +\tan\phi}{1-\tan\theta\tan\phi}
  4. \cot(\theta+\phi) = \frac{\cot\theta\cot\phi-1}{\cot\theta+\cot\phi}

Problem set

In each of the following problems, find the values of \sin 2\theta and \cos 2\theta using the information given.

  1. \sin\theta = \frac{1}{5}
  2. \cos\theta = \frac{1}{3}
  3. \sin\theta = \frac{1}{10}
  4. \cos\theta = \frac{2}{7}

Problem set

Using the formulas for \sin\left(\frac{\theta}{2}\right) and \cos\left(\frac{\theta}{2}\right), find the values of the following.

  1. \cos\left(15^\circ\right)
  2. \sin\left(\frac{\pi}{8}\right)
  3. \tan\left(15^\circ\right)
  4. \cot\left(\frac{\pi}{8}\right)

Problem set

  1. Express \sin(3\theta) is terms of \sin\theta alone.
  2. Express \cos(3\theta) is terms of \cos\theta alone.
  3. Express \cos(4\theta) is terms of \cos\theta alone.
  4. Express \cos(4\theta) is terms of \sin\theta alone.

Laws of sines and cosines

Exercises

Problem set

In each of the following problems, certain angle measures and side lengths are given for a triangle ABC. Find the side lengths and angle measures that are not given.

  1. \angle A = \frac{\pi}{6},\,\, \angle B = \frac{\pi}{6},\,\, AB = 2
  2. \angle C = 75^\circ,\,\, \angle B = 45^\circ,\,\, BA = 3
  3. \angle B = 105^\circ,\,\, BC = \sqrt{2},\,\, AB = 1
  4. \angle A = \frac{2\pi}{3},\,\, AC = \sqrt{6},\,\, CB = 3
  5. \angle C = \frac{5\pi}{6},\,\, AB = 1,\,\, BC = \sqrt{2}
  6. \angle B = \frac{\pi}{4},\,\, AC = \sqrt{2},\,\, CB = 4
  7. \angle A = \frac{\pi}{3},\,\, BC = \sqrt{2},\,\, CA = \frac{2}{\sqrt{3}}
  8. \angle C = \frac{\pi}{6},\,\, AB = \sqrt{2},\,\, BC = 2

Trigonometric functions

Graphing

Exercises

Problem set
  1. Evaluate: \sin x for x = -2\pi,-\frac{7\pi}{4}, -\frac{3\pi}{2}, -\frac{5\pi}{4}, -\pi, -\frac{3\pi}{4}, -\frac{\pi}{2}, -\frac{\pi}{4}, 0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi, \frac{5\pi}{4}, \frac{3\pi}{2}, \frac{7\pi}{4}, 2\pi. Using the values as a guide, plot \sin x on the coordinate plane for -2\pi \le x \le 2\pi.
  2. Evaluate: \cos x for x = -2\pi,-\frac{7\pi}{4}, -\frac{3\pi}{2}, -\frac{5\pi}{4}, -\pi, -\frac{3\pi}{4}, -\frac{\pi}{2}, -\frac{\pi}{4}, 0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi, \frac{5\pi}{4}, \frac{3\pi}{2}, \frac{7\pi}{4}, 2\pi. Using the values as a guide, plot \cos x on the coordinate plane for -2\pi \le x \le 2\pi.
Problem set

For each of the following, graph the function.

  1. \sin(x) + 2
  2. \cos(x) - 3
  3. 2\sin x
  4. 0.8\cos x
  5. 2\sin x - 3
  6. 0.5\cos x - 1.5
  7. 1.5\sin x - 2.5
Problem set

For each of the following, graph the function.

  1. 2\sin\left(x-3\right)
  2. 3\cos\left(x+10\right)
  3. 5\sin\left(x+4\right)-7
  4. 7\sin\left(x-5\right)+2
  5. 10\cos\left(x-5\right)+5
  6. 7\cos\left(x+3\right)-5
Problem set

For each of the following, graph the function.

  1. \sin\left(5x\right)
  2. 5\cos\left(3x\right)-2
  3. 3\sin\left(4x\right)-2
  4. \cos\left(3\left(x+20\right)\right)
  5. \sin\left(2\left(x-50\right)\right)
  6. 3\cos\left(0.5(x-10)\right)
  7. 0.5\sin\left(2(x-15)\right)+2.5
Problem set

For each of the following, graph the function, identify its domain and range, and find the period, amplitude, frequency, mid-line and horizontal(phase) shift.

  1. \sin\left(2\left(x-20\right)\right)
  2. \cos\left(3\left(x+50\right)\right)
  3. \sin\left(x-\frac{\pi}{2}\right)+2
  4. 3\cos\left(x-10\right)-3
  5. 0.5\sin\left(2x\right)-2.5
  6. 3\cos\left(2\left(x-40\right)\right)-20
  7. 0.5\sin\left(0.5\left(x+20\right)\right)-0.5
  8. 7\sin\left(\frac{x-10}{2}\right)-2
  9. 3\cos\left(2x-5\right)-5
  10. 5\sin\left(3x-4\right)-7
  11. 10\cos\left(\frac{x}{2}-\frac{\pi}{2}\right)-7
Problem set

For each of the following graphs, give the expression for the function.

Problem set

For each of the following, graph the function.

  1. \tan\left(x+\frac{\pi}{2}\right)
  2. \tan\left(2x\right)
  3. \tan\left(x+\pi\right)
  4. \cot\left(x-\frac{\pi}{2}\right)
  5. \cot\left(\frac{x}{2}\right)
  6. \cot\left(x-\pi\right)
Problem set

For each of the following, graph the function.

  1. \csc\left(x+\frac{\pi}{2}\right)
  2. \csc\left(2x\right)
  3. \sec\left(x-\frac{\pi}{2}\right)
  4. \sec\left(\frac{x}{2}\right)
Problem set

For each of the following, graph the function.

  1. 2x + 3 + \sin(x)
  2. x^2 + \cos(2x)
  3. x^3 + \sin\left(x-\frac{\pi}{2}\right)
  4. x\cos\left(x\right)
  5. x^2\sin\left(x\right)
  6. e^x\cos\left(x\right)
  7. \frac{1}{x}\cos\left(x\right)
  8. 2^{-x}\sin\left(x\right)
Problem set

Determine restrictions on the domains where the following functions have inverses. Plot the functions and their inverses on the restricted domains. Identify the domain and range of the inverse functions.

  1. \sin x
  2. \cos x
  3. \tan x
  4. \cot x
  5. \csc x
  6. \sec x

Applications

Exercises

Problem set

Assume that an ant is on the rim of a wheel of a bike standing on the ground, and that the ant starts off from its initial position and travels along the rim at a uniform speed. We denote the the radius (in feet) of the wheel by R, the initial position of the ant by I, the time (in seconds) it takes to complete one cycle around the rim by P and the direction in which the ant travels by D.

For each of the following variations, give an expression that models the height of the ant from the ground as a function of time.

  1. R = 1, P = 2\pi, I = \mbox{right-end of a horizontal spoke of the wheel}, D = \mbox{counter-clockwise}
  2. R = 1, P = \pi, I = \mbox{right-end of a horizontal spoke of the wheel}, D = \mbox{counter-clockwise}
  3. R = 1, P = 4\pi, I = \mbox{right-end of a horizontal spoke of the wheel}, D = \mbox{counter-clockwise}
  4. R = 1, P = 2\pi, I = \mbox{top-end of a vertical spoke of the wheel}, D = \mbox{counter-clockwise}
  5. R = 1, P = 2\pi, I = \mbox{left-end of a horizontal spoke of the wheel}, D = \mbox{counter-clockwise}
  6. R = 1, P = \pi, I = \mbox{top-end of a vertical spoke of the wheel}, D = \mbox{counter-clockwise}
  7. R = 1, P = 4\pi, I = \mbox{left-end of a horizontal spoke of the wheel}, D = \mbox{counter-clockwise}
  8. R = 2, P = 2\pi, I = \mbox{right-end of a horizontal spoke of the wheel}, D = \mbox{counter-clockwise}
  9. R = 2, P = \pi, I = \mbox{right-end of a horizontal spoke of the wheel}, D = \mbox{counter-clockwise}
  10. R = \frac{1}{2}, P = \frac{\pi}{2}, I = \mbox{bottom-end of a vertical spoke of the wheel}, D = \mbox{counter-clockwise}
  11. R = 1, P = 2\pi, I = \mbox{right-end of a horizontal spoke of the wheel}, D = \mbox{clockwise}
  12. R = 1, P = 8\pi, I = \mbox{right-end of a horizontal spoke of the wheel}, D = \mbox{clockwise}
  13. R = 3, P = \pi, I = \mbox{bottom-end of a vertical spoke of the wheel}, D = \mbox{clockwise}
  14. R = \frac{1}{4}, P = \frac{\pi}{4}, I = \mbox{left-end of a horizontal spoke of the wheel}, D = \mbox{clockwise}
  15. R = \frac{1}{2}, P = \frac{\pi}{2}, I = \mbox{top-end of a vertical spoke of the wheel}, D = \mbox{clockwise}
  16. R = 5, P = 15, I = \mbox{left-end of a vertical spoke of the wheel}, D = \mbox{counter-clockwise}
  17. R = 7, P = 11, I = \mbox{top-end of a vertical spoke of the wheel}, D = \mbox{clockwise}
Problem set
  1. A bicycle has a wheel of radius 1^\prime. A particle of dust is at the right end of a horizontal spoke of the front bicycle wheel. The bicycle is moving backwards at a speed of 1 feet/sec. Give an expression to model the height of the dust particle from the ground as a function of time.
  2. A Ferris wheel 40^\prime in diameter completes one revolution in 2\pi seconds. The bottom of the Ferris wheel is 4^\prime above the ground. Give an expression to model the height of a rider as a function of time, assuming the rider boards in the bottom most cabin of the Ferris wheel.
  3. A Ferris wheel 100^\prime in diameter completes one revolution in 2 minutes. The bottom of the Ferris wheel is 5^\prime above the ground. Give an expression to model the height of a rider as a function of time(in seconds), assuming the rider boards in the bottom most cabin of the Ferris wheel.
  4. Saahas is skipping rocks at a lake. As a rock hit the surface of the water, a wave formed and radiated outward. If the peak of the wave is 1.2'' above the surface and the radial distance between two peaks is 3'', give a trigonometric expression to approximate the height of the water as a function of radial distance from where the wave formed. Assume the wave has a low at the point where the rock hits the water.
Problem set
  1. As Jaden pumps air into his basketball using his new bike pump, the volume of air inside the pump changes between 10 cubic centimeters and 300 cubic centimeters. Jaden takes 3 seconds for each stroke. Give a trigonometric expression to approximate the volume of air as a function of time. Assume time is measured from when the air is at a minimum inside the pump.
  2. In a particular month at Ocean city, Maryland, the high-tide and low-tide followed a particular pattern. Every day, the high-tide rose to a height of 1.5m at midnight, while the low-tide rose to 0.8m at noon. Give a trigonometric expression to approximate the height of the tide as a function of time (in hours) elapsed from:
    1. 6PM of a certain day
    2. midnight of a certain day
    3. 6AM of a certain day
    4. 4AM of a certain day
  3. In Washington DC, the longest day of the year 2020 was Jun 20th, with a day-length of 14 hours, 54 minutes, while the shortest of the year occurred half-year later with a day-length of 9 hours, 27 minutes. Give a trigonometric expression to approximately estimate the day-length (in minutes) as a function of number of days elapsed from:
    1. Jun 20th, 2020
    2. Jan 1st, 2020. (Note: Number of days between Jan 1st and Jun 20th was 171 days.)
  4. For a particular police car, as its siren is turned on, the sound goes up from 70 decibels to 100 decibels in 90 seconds, then down from 100 decibels to 70 decibels in the next 90 seconds, then again up from 70 decibels to 100 decibels in the next 90 seconds and so on. Give a trigonometric expression to approximate the level of sound as a function of time elapsed from turning the siren on.
  5. An ideal horizontal spring of length 1^\prime is anchored on one end to a vertical wall, while a movable mass is attached to the other end of the spring. The spring and the mass are initially at rest. A force is applied to displace the mass by 5'' from its resting position. The restoring force in the spring causes the mass to oscillate back and forth passing through its resting position. The time it takes for the mass to go from the right extreme to the left extreme is 4 seconds. Give a trigonometric expression for:
    1. the distance of the mass from the wall as a function of time
    2. the displacement of the mass from its resting position as a function of time