Learn – Geometry – Trigonometry

1 Similar triangles

1.1 Exercises

1.1.1 Problem set

In each of the following problems, \triangle ABC \sim \triangle DEF. The side lengths of \triangle ABC are given, and one side length of \triangle DEF is given. Find the missing side lengths of \triangle DEF.
















2 Special triangles

2.1 Exercises

2.1.1 Problem set

For the following problems, assume ABC is a right triangle with \angle B = 90^\circ. For each problem, given the measures of one angle and one side, find the measures of the other two sides of the triangle. Do not use trigonometric ratios to solve these.

  1. \angle C = 45^\circ, AB = 2
  2. \angle A = 45^\circ, BC = 3
  3. \angle C = 45^\circ, AB = 4
  4. \angle A = 45^\circ, AC = 2
  5. \angle C = 45^\circ, AC = 6

2.1.2 Problem set

For the following problems, assume ABC is a right triangle with \angle B = 90^\circ. For each problem, given the measures of one angle and one side, find the measures of the other two sides of the triangle. Do not use trigonometric ratios to solve these.

  1. \angle C = 45^\circ, AC = 20
  2. \angle A = 30^\circ, BC = 2
  3. \angle C = 60^\circ, BC = 50
  4. \angle A = 30^\circ, AB = 5\sqrt{3}
  5. \angle C = 30^\circ, BC = 10\sqrt{3}
  6. \angle A = 60^\circ, AC = 10
  7. \angle C = 60^\circ, AC = 8

2.1.3 Problem set

For the following problems, assume ABC is a right triangle with \angle B = 90^\circ. For each problem, given the measures of one angle and one side, find the measures of the other two sides of the triangle. Do not use trigonometric ratios to solve these.

  1. \angle A = 45^\circ, AC = 100
  2. \angle A = 45^\circ, BC = \sqrt{6}
  3. \angle A = 30^\circ, AB = 12
  4. \angle C = 60^\circ, AB = 9
  5. \angle C = 60^\circ, BC = 4\sqrt{3}
  6. \angle C = 30^\circ, BC = 4
  7. \angle A = 60^\circ, AC = \sqrt{6}

3 Definitions of ratios in right triangle context

3.1 Exercises

3.1.1 Problem set

In each of the following problems, find \sin\theta, \cos\theta, \tan\theta using the figure given.

  1. \theta = 49^\circ


  2. \theta = 65^\circ


  3. \theta = 26.49^\circ


  4. \theta = 53^\circ


3.1.2 Problem set

Find the values of the following using a geometry toolbox or geometry software. Do not use a calculator.

  1. \sin 35^\circ
  2. \sin 53^\circ
  3. \cos 25^\circ
  4. \cos 41^\circ
  5. \tan 65^\circ
  6. \tan 20^\circ

3.1.3 Problem set

In each of the following problems, assume we have a right triangle ABC with \angle ABC = 90^\circ and \angle BAC = \theta.

  1. Find \sin \theta, if BC = 3, AB = 4.
  2. Find \cos \theta, if BC = 4, AC = 5.
  3. Find \tan \theta, if BC = 3, AC = 5.
  4. Find \sin \theta, if AB = 6, AC = 10.
  5. Find \sin \theta, if BC = 1, AB = 1.
  6. Find \tan \theta, if BC = \sqrt{3}, AC = 2.

3.1.4 Problem set

In each of the following problems, assume we have a right triangle ABC with \angle ABC = 90^\circ and \angle BAC = \theta.

  1. Find \sin \theta, if BC = 2, AB = 2\sqrt{3}.
  2. Find \cos \theta, if BC = 3, AB = 3.
  3. Find \cos \theta, if BC = 9, AB = 3\sqrt{3}.
  4. Find \tan \theta, if AB = \sqrt{3}, AC = 2.
  5. Find \tan \theta, if AB = \frac{1}{2}, AC = 1.

3.1.5 Problem set

In each of the following problems, assume we have a right triangle ABC with \angle ABC = 90^\circ and \angle BAC = \theta.

  1. Find \csc \theta, if BC = 3, AB = 4.
  2. Find \sec \theta, if BC = 8, AB = 6.
  3. Find \csc \theta, if BC = 3, AB = 3.
  4. Find \cot \theta, if BC = 6, AC = 10.
  5. Find \sec \theta, if BC = 1, AC = 2.
  6. Find \cot \theta, if BC = \sqrt{3}, AC = 2.

3.1.6 Problem set

For the following problems, assume ABC is a right triangle with \angle B = 90^\circ. For each problem, given the measures of one angle and one side, find the measures of the other two sides of the triangle.

  1. \angle C = 23^\circ, AB = 2
  2. \angle A = 71^\circ, AC = 2
  3. \angle C = 35^\circ, AC = 4
  4. \angle A = 80^\circ, BC = 2
  5. \angle A = 18^\circ, AB = 2

4 Ratios of complimentary angles

4.1 Exercises

4.1.1 Problem set

In each of the following problems, \theta is one of the acute angles in a right triangle.

  1. \sin \theta = \frac{4}{5}. Find \cos (90^\circ - \theta).
  2. \cos \theta = \frac{5}{13}. Find \sin (90^\circ - \theta).
  3. \tan \theta = \frac{5}{12}. Find \cot (90^\circ - \theta).
  4. \sec \theta = \frac{5}{3}. Find \csc (90^\circ - \theta).
  5. \csc \theta = \frac{13}{5}. Find \sec (90^\circ - \theta).

4.1.2 Problem set

In each of the following problems, \theta is one of the acute angles in a right triangle.

  1. \sin \theta = \frac{4}{5}. Find \tan (90^\circ - \theta).
  2. \cos \theta = \frac{5}{13}. Find \cot (90^\circ - \theta).
  3. \tan \theta = \frac{5}{12}. Find \sec (90^\circ - \theta).
  4. \sec (90^\circ - \theta) = \frac{5}{3}. Find \tan \theta.
  5. \csc (90^\circ - \theta) = \frac{13}{5}. Find \cot \theta.

5 Definitions of ratios in unit circle context

5.1 Exercises

5.1.1 Motivation

5.1.1.1 Problem set
  1. In the following diagram, a circle is drawn with its center at the origin of a coordinate plane, and a radius making angle \theta with X-axis is shown. Find the values of \sin \theta, \cos \theta and \tan \theta.
  2. In the following diagram, a circle is drawn with its center at the origin of a coordinate plane, and a radius making angle \theta with X-axis is shown. Find the values of \sin \theta, \cos \theta and \tan \theta.

5.1.2 Angles between 0 degrees and 360 degrees

5.1.2.1 Problem set

Find the values of the following.

  1. \sin (0^\circ)
  2. \cos (90^\circ)
  3. \cos (180^\circ)
  4. \sin (270^\circ)
  5. \tan (180^\circ)
  6. \tan (360^\circ)
5.1.2.2 Problem set

Find the values of the following.

  1. \sin (135^\circ)
  2. \cos (225^\circ)
  3. \sin (120^\circ)
  4. \cos (315^\circ)
  5. \sin (210^\circ)
5.1.2.3 Problem set

Find the values of the following.

  1. \sec (315^\circ)
  2. \tan (120^\circ)
  3. \cot (240^\circ)
  4. \csc (300^\circ)
5.1.2.4 Problem set

Find the values of the following.

  1. \sin \left(\frac{\pi}{4}\right)
  2. \cos \left(\frac{\pi}{6}\right)
  3. \tan \left(\frac{\pi}{3}\right)
  4. \sin \left(\frac{\pi}{2}\right)
  5. \cos \left(\frac{3\pi}{2}\right)
  6. \tan \left(\pi\right)
5.1.2.5 Problem set

Evaluate the following

  1. \cot \left(\frac{3\pi}{4}\right)
  2. \sin \left(\frac{7\pi}{4}\right)
  3. \csc \left(\frac{11\pi}{6}\right)
  4. \sec \left(\frac{5\pi}{3}\right)

5.1.3 Angles greater than 360 degrees

5.1.3.1 Problem set

Evaluate the following.

  1. \cos (450^\circ)
  2. \sin (540^\circ)
  3. \tan (720^\circ)
5.1.3.2 Problem set

Evaluate the following.

  1. \sin (405^\circ)
  2. \cos (495^\circ)
  3. \sin (390^\circ)
  4. \tan (510^\circ)
  5. \csc (585^\circ)
  6. \sec (675^\circ)
  7. \cot (600^\circ)
5.1.3.3 Problem set

Express each of the following angles as M \pm \theta, where M is a multiple of 2\pi and 0 \le \theta \le \pi. For example, \frac{19\pi}{6} = \frac{24\pi}{6} - \frac{5\pi}{6} = 4\pi - \frac{5\pi}{6}. Then, show a line segment making the specified angle with positive X-axis of a coordinate plane.

  1. \frac{7\pi}{3}
  2. \frac{13\pi}{6}
  3. \frac{9\pi}{4}
  4. \frac{7\pi}{4}
  5. \frac{11\pi}{6}
  6. \frac{5\pi}{3}
5.1.3.4 Problem set

Express each of the following angles as M \pm \theta, where M is a multiple of 2\pi and 0 \le \theta \le \pi. For example, \frac{19\pi}{6} = \frac{24\pi}{6} - \frac{5\pi}{6} = 4\pi - \frac{5\pi}{6}. Then, show a line segment making the specified angle with positive X-axis of a coordinate plane.

  1. \frac{11\pi}{4}
  2. \frac{8\pi}{3}
  3. \frac{17\pi}{6}
  4. \frac{15\pi}{4}
  5. \frac{11\pi}{3}
5.1.3.5 Problem set

Express each of the following angles as M \pm \theta, where M is a multiple of 2\pi and 0 \le \theta \le \pi. For example, \frac{19\pi}{6} = \frac{24\pi}{6} - \frac{5\pi}{6} = 4\pi - \frac{5\pi}{6}. Then, show a line segment making the specified angle with positive X-axis of a coordinate plane.

  1. \frac{35\pi}{4}
  2. \frac{32\pi}{3}
  3. \frac{65\pi}{6}
  4. \frac{47\pi}{4}
  5. \frac{46\pi}{3}
  6. \frac{67\pi}{6}
5.1.3.6 Problem set

Express each of the following angles as M \pm \theta, where M is a multiple of 2\pi and 0 \le \theta \le \pi. For example, \frac{19\pi}{6} = \frac{24\pi}{6} - \frac{5\pi}{6} = 4\pi - \frac{5\pi}{6}. Then, show a line segment making the specified angle with positive X-axis of a coordinate plane.

  1. \frac{13\pi}{3}
  2. \frac{15\pi}{4}
  3. \frac{75\pi}{20}
5.1.3.7 Problem set

Evaluate the following

  1. \sin \left(2\pi + \frac{\pi}{3}\right)
  2. \cos \left(2\pi + \frac{\pi}{4}\right)
  3. \tan \left(2\pi + \frac{\pi}{6}\right)
  4. \csc \left(2\pi + \frac{2\pi}{3}\right)
  5. \sec \left(2\pi + \frac{3\pi}{4}\right)
  6. \csc \left(2\pi - \frac{\pi}{3}\right)
  7. \sec \left(2\pi - \frac{\pi}{6}\right)
  8. \cot \left(2\pi - \frac{3\pi}{4}\right)
5.1.3.8 Problem set

Evaluate the following

  1. \sin \left(4\pi + \frac{5\pi}{6}\right)
  2. \cos \left(20\pi + \frac{3\pi}{4}\right)
  3. \tan \left(40\pi - \frac{2\pi}{3}\right)
  4. \csc \left(24\pi - \frac{5\pi}{6}\right)
  5. \sec \left(30\pi - \frac{3\pi}{4}\right)
5.1.3.9 Problem set

Evaluate the following

  1. \sin \left(122\pi\right)
  2. \sec \left(29\pi\right)
  3. \csc \left(1000\pi+\frac{\pi}{2}\right)
  4. \sin \left(77\pi+\frac{\pi}{4}\right)
  5. \tan \left(101157\pi+\frac{\pi}{6}\right)
  6. \tan \left(101157\pi-\frac{\pi}{6}\right)
5.1.3.10 Problem set

Evaluate the following

  1. \csc \left(\frac{14\pi}{3}\right)
  2. \tan \left(\frac{19\pi}{4}\right)
  3. \sec \left(\frac{31\pi}{6}\right)
  4. \cot \left(\frac{62\pi}{3}\right)
5.1.3.11 Problem set

Evaluate the following

  1. \tan (960^\circ)
  2. \sec (1380^\circ)
  3. \csc (945^\circ)
  4. \cot (870^\circ)
  5. \sec (900^\circ)

5.1.4 Negative angles

5.1.4.1 Problem set

Evaluate the following

  1. \sin (-30^\circ)
  2. \cos (-60^\circ)
  3. \tan (-135^\circ)
  4. \cos (-315^\circ)
  5. \sec (-495^\circ)
  6. \csc (-675^\circ)
  7. \cot (-1500^\circ)
  8. \sec (-1290^\circ)
  9. \csc (-810^\circ)
5.1.4.2 Problem set

Evaluate the following

  1. \cot \left(-\frac{\pi}{4}\right)
  2. \csc \left(-\frac{\pi}{6}\right)
  3. \sec \left(-\frac{\pi}{3}\right)
  4. \csc \left(-\frac{\pi}{2}\right)
  5. \sec \left(-2\pi\right)
  6. \cot \left(-\pi\right)
5.1.4.3 Problem set

Evaluate the following

  1. \csc \left(-\frac{2\pi}{3}\right)
  2. \cos \left(-\frac{5\pi}{4}\right)
  3. \sec \left(-\frac{5\pi}{6}\right)
5.1.4.4 Problem set

Express each of the following angles as M \pm \theta, where M is a multiple of 2\pi and 0 \le \theta \le \pi. For example, \frac{19\pi}{6} = \frac{24\pi}{6} - \frac{5\pi}{6} = 4\pi - \frac{5\pi}{6}. Then, show a line segment making the specified angle with positive X-axis of a coordinate plane.

  1. -\frac{7\pi}{3}
  2. -\frac{13\pi}{6}
  3. -\frac{7\pi}{4}
  4. -\frac{11\pi}{6}
  5. -\frac{8\pi}{3}
5.1.4.5 Problem set

Express each of the following angles as M \pm \theta, where M is a multiple of 2\pi and 0 \le \theta \le \pi. For example, \frac{19\pi}{6} = \frac{24\pi}{6} - \frac{5\pi}{6} = 4\pi - \frac{5\pi}{6}. Then, show a line segment making the specified angle with positive X-axis of a coordinate plane.

  1. -\frac{11\pi}{3}
  2. -\frac{83\pi}{4}
  3. -\frac{62\pi}{3}
  4. -\frac{37\pi}{6}
  5. -\frac{45\pi}{4}
5.1.4.6 Problem set

Express each of the following angles as M \pm \theta, where M is a multiple of 2\pi and 0 \le \theta \le \pi. For example, \frac{19\pi}{6} = \frac{24\pi}{6} - \frac{5\pi}{6} = 4\pi - \frac{5\pi}{6}. Then, show a line segment making the specified angle with positive X-axis of a coordinate plane.

  1. -\frac{25\pi}{6}
  2. -\frac{140\pi}{60}
  3. -\frac{17\pi}{6}
  4. -\frac{14\pi}{3}
5.1.4.7 Problem set

Evaluate the following

  1. \tan \left(-50\pi\right)
  2. \sec \left(-1001\pi\right)
  3. \cot \left(-1001\pi+\frac{\pi}{2}\right)
  4. \sec \left(-8123\pi+\frac{\pi}{6}\right)
  5. \tan \left(-77\pi+\frac{\pi}{3}\right)
5.1.4.8 Problem set

Evaluate the following

  1. \tan \left(-2324\pi+\frac{\pi}{4}\right)
  2. \cot \left(-4127\pi-\frac{\pi}{4}\right)
  3. \sec \left(-1078\pi-\frac{\pi}{3}\right)
  4. \csc \left(-3333\pi+\frac{\pi}{3}\right)
5.1.4.9 Problem set

Evaluate the following

  1. \sin \left(-\frac{33\pi}{6}\right)
  2. \sec \left(-\frac{29\pi}{6}\right)
  3. \cot \left(-\frac{19\pi}{4}\right)
  4. \csc \left(\frac{251\pi}{2}\right)
  5. \sec \left(\frac{-511\pi}{2}\right)

6 Miscellaneous

6.1 Exercises

6.1.1 Problem set

In each of the following problems, determine the value of \theta, if \theta is one of the acute angles in a right triangle.

  1. \sin \theta = \cos \theta
  2. \tan \theta = \cot \theta
  3. 2\tan \theta = \sec \theta
  4. \sec \theta = \sqrt{3}\csc \theta

6.1.2 Problem set

For the following problems, assume 0^\circ < \theta < 90^\circ.

  1. Express \tan\theta in terms of \sin\theta and \cos\theta.
  2. Express \cot\theta in terms of \sin\theta and \cos\theta.
  3. Express \sin\theta in terms of \cot\theta and \cos\theta.
  4. Express \cos\theta in terms of \cot\theta and \sin\theta.
  5. Express \cot\theta in terms of \sec\theta and \cos\left(90^\circ-\theta\right).
  6. Express \sec\theta in terms of \sin\theta and \tan\left(90^\circ-\theta\right).
  7. Express \csc\theta in terms of \tan\theta and \sin\left(90^\circ-\theta\right).

6.1.3 Problem set

Find the reference angle of each of the following.

  1. 210^\circ
  2. -300^\circ
  3. -360^\circ\times 2 - 45^\circ
  4. 180^\circ\times 5 - 30^\circ
  5. -180^\circ\times 5 + 60^\circ
  6. 480^\circ
  7. -855^\circ
  8. -1110^\circ

6.1.4 Problem set

For each of the following, find the reference angle for the angle involved. Then, use the reference angle and the ASTC mnemonic to evaluate the trigonometric ratio.

  1. \cos (495^\circ)
  2. \tan (960^\circ)
  3. \csc (1380^\circ)
  4. \sin (-480^\circ)
  5. \sec (-960^\circ)
  6. \cot (-1395^\circ)

6.1.5 Problem set

Find the reference angle of each of the following.

  1. 119\pi+\frac{\pi}{6}
  2. 119\pi-\frac{\pi}{6}
  3. 2198\pi-\frac{\pi}{3}
  4. 2198\pi+\frac{\pi}{3}
  5. \frac{7\pi}{4}
  6. -\frac{19\pi}{4}
  7. \frac{31\pi}{6}
  8. -\frac{14\pi}{3}

6.1.6 Problem set

In the following, assume 0 < \theta < \frac{\pi}{2}. There are two blanks to be filled on the right hand side of each of the problems. Fill in the blanks appropriately. Use + or - for the first blank, and one of the six trigonometric ratios (\sin\theta, \cos\theta, \tan\theta, \csc\theta,\sec\theta,\cot\theta) for the second blank. For example, if \cos(-\theta) = \underline{\hspace{0.5cm}}\,\,\underline{\hspace{1cm}}, the first blank takes + and the second blank takes \cos\theta.

  1. \sin(\frac{\pi}{2} + \theta) = \underline{\hspace{0.5cm}}\,\,\underline{\hspace{1cm}}
  2. \sec(\pi + \theta) = \underline{\hspace{0.5cm}}\,\,\underline{\hspace{1cm}}
  3. \csc\left(-\frac{3\pi}{2} + \theta\right) = \underline{\hspace{0.5cm}}\,\,\underline{\hspace{1cm}}
  4. \cot\left(\frac{3\pi}{2} + \theta\right) = \underline{\hspace{0.5cm}}\,\,\underline{\hspace{1cm}}
  5. \tan(-8\pi - \theta) = \underline{\hspace{0.5cm}}\,\,\underline{\hspace{1cm}}
  6. \tan(-\theta) = \underline{\hspace{0.5cm}}\,\,\underline{\hspace{1cm}}
  7. \sin(-\theta) = \underline{\hspace{0.5cm}}\,\,\underline{\hspace{1cm}}

6.1.7 Problem set

In the following figure, \angle BAD = \theta and \angle DAC = \varphi. Line segments CE, GD, CD and DF are constructed such that \angle AEC, \angle CGD, \angle ADC and \angle AFD are all right angles. And, AC = 1 unit.

The following problems are based on the above figure.

  1. Which of the following is \sin(\theta+\varphi) equal to: CD, AE, FD or CE?
  2. Which of the following is AD equal to: \sin\varphi, \cos\varphi, \sin\theta or \cos\theta?
  3. Using right triangle AFD, express FD in terms of \sin\theta and \cos\varphi. Hint: Use problem 2.
  4. Which of the following is CD equal to: \sin\varphi, \cos\varphi, \sin\theta or \cos\theta?
  5. Express \angle CHD in terms of \theta.
  6. Express \angle HCD in terms of \theta.
  7. Using right triangle CGD, express CG in terms of \sin\varphi and \cos\theta. Hint: Use problems 4 and 6.
  8. Express \sin(\theta+\varphi) in terms of \sin\theta, \cos\theta, \sin\varphi and \cos\varphi. Hint: Use problems 1, 3 and 7.

6.1.8 Problem set

  1. For 810^\circ < \theta < 900^\circ, what are the minimum and maximum values of \sin \theta, \cos \theta and \tan \theta?
  2. For -810^\circ < \theta < -720^\circ, what are the minimum and maximum values of \sin \theta, \cos \theta and \tan \theta?
  3. For -1620^\circ \le \theta < -1575^\circ, find the minimum and maximum values of \sin \theta and \cos \theta.
  4. For 810^\circ < \theta \le 840^\circ, find the minimum and maximum values of \tan \theta.
  5. For 0^\circ < \theta \le 90^\circ, find the minimum and maximum values of \csc \theta.
  6. For 90^\circ < \theta \le 180^\circ, find the minimum and maximum values of \sec \theta.
  7. For -90^\circ \le \theta < 0^\circ, find the minimum and maximum values of \cot \theta.

6.1.9 Problem set

  1. For -945^\circ < \theta < -900^\circ, find the minimum and maximum values of \sec \theta.
  2. For 810^\circ < \theta < 870^\circ, find the minimum and maximum values of \cot \theta.
  3. For 1620^\circ < \theta < 1665^\circ, find the minimum and maximum values of \csc \theta.
  4. For -90^\circ < \theta < 90^\circ, find the minimum and maximum values of \tan \theta.
  5. For -405^\circ < \theta < -300^\circ, find the minimum and maximum values of \cos \theta.
  6. For -510^\circ < \theta < -405^\circ, find the minimum and maximum values of \sin \theta.
  7. For -690^\circ < \theta < -570^\circ, find the minimum and maximum values of \csc \theta.
  8. Say, 0^\circ < \theta < 45^\circ. Explain the minimum and maximum possible values of \sin \theta, \cos \theta and \tan \theta?
  9. Prove: \sin^2 \theta + \cos^2 \theta = 1
  10. Prove: \sec^2 \theta - \tan^2 \theta = 1