Learn – Geometry – Trigonometry

1 Similar triangles
- 1.1 Exercises
  - 1.1.1 Problem set
2 Special triangles
- 2.1 Exercises
3 Definitions of ratios in right triangle context
- 3.1 Exercises
4 Ratios of complimentary angles
- 4.1 Exercises
  - 4.1.1 Problem set
  - 4.1.2 Problem set
5 Definitions of ratios in unit circle context
- 5.1 Exercises
6 Miscellaneous
- 6.1 Exercises

Similar triangles

Exercises

Problem set

$\triangle ABC \sim \triangle DEF$

$\triangle ABC$

$\triangle DEF$

Special triangles

Exercises

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.

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

Problem set

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

Problem set

$\angle A = 45^\circ, AC = 100$
$\angle A = 45^\circ, BC = \sqrt{6}$
$\angle A = 30^\circ, AB = 12$
$\angle C = 60^\circ, AB = 9$
$\angle C = 60^\circ, BC = 4\sqrt{3}$
$\angle C = 30^\circ, BC = 4$
$\angle A = 60^\circ, AC = \sqrt{6}$

Definitions of ratios in right triangle context

Exercises

Problem set

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

$\theta = 49^\circ$

$\theta = 65^\circ$

$\theta = 26.49^\circ$

$\theta = 53^\circ$

Problem set

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

$\sin 35^\circ$
$\sin 53^\circ$
$\cos 25^\circ$
$\cos 41^\circ$
$\tan 65^\circ$
$\tan 20^\circ$

Problem set

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

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

Problem set

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

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

Problem set

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

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

Problem set

$\angle C = 23^\circ, AB = 2$
$\angle A = 71^\circ, AC = 2$
$\angle C = 35^\circ, AC = 4$
$\angle A = 80^\circ, BC = 2$
$\angle A = 18^\circ, AB = 2$

Ratios of complimentary angles

Exercises

Problem set

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

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

Problem set

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

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

Definitions of ratios in unit circle context

Exercises

Motivation

Problem set

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 $\tan \theta$ .
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 $\tan \theta$ .

Angles between 0 degrees and 360 degrees

Problem set

Find the values of the following.

$\sin (0^\circ)$
$\cos (90^\circ)$
$\cos (180^\circ)$
$\sin (270^\circ)$
$\tan (180^\circ)$
$\tan (360^\circ)$

Problem set

Find the values of the following.

$\sin (135^\circ)$
$\cos (225^\circ)$
$\sin (120^\circ)$
$\cos (315^\circ)$
$\sin (210^\circ)$

Problem set

Find the values of the following.

$\sec (315^\circ)$
$\tan (120^\circ)$
$\cot (240^\circ)$
$\csc (300^\circ)$

Problem set

Find the values of the following.

$\sin \left(\frac{\pi}{4}\right)$
$\cos \left(\frac{\pi}{6}\right)$
$\tan \left(\frac{\pi}{3}\right)$
$\sin \left(\frac{\pi}{2}\right)$
$\cos \left(\frac{3\pi}{2}\right)$
$\tan \left(\pi\right)$

Problem set

Evaluate the following

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

Angles greater than 360 degrees

Problem set

Evaluate the following.

$\cos (450^\circ)$
$\sin (540^\circ)$
$\tan (720^\circ)$

Problem set

Evaluate the following.

$\sin (405^\circ)$
$\cos (495^\circ)$
$\sin (390^\circ)$
$\tan (510^\circ)$
$\csc (585^\circ)$
$\sec (675^\circ)$
$\cot (600^\circ)$

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.

$\frac{7\pi}{3}$
$\frac{13\pi}{6}$
$\frac{9\pi}{4}$
$\frac{7\pi}{4}$
$\frac{11\pi}{6}$
$\frac{5\pi}{3}$

Problem set

$\frac{11\pi}{4}$
$\frac{8\pi}{3}$
$\frac{17\pi}{6}$
$\frac{15\pi}{4}$
$\frac{11\pi}{3}$

Problem set

$\frac{35\pi}{4}$
$\frac{32\pi}{3}$
$\frac{65\pi}{6}$
$\frac{47\pi}{4}$
$\frac{46\pi}{3}$
$\frac{67\pi}{6}$

Problem set

$\frac{13\pi}{3}$
$\frac{15\pi}{4}$
$\frac{75\pi}{20}$

Problem set

Evaluate the following

$\sin \left(2\pi + \frac{\pi}{3}\right)$
$\cos \left(2\pi + \frac{\pi}{4}\right)$
$\tan \left(2\pi + \frac{\pi}{6}\right)$
$\csc \left(2\pi + \frac{2\pi}{3}\right)$
$\sec \left(2\pi + \frac{3\pi}{4}\right)$
$\csc \left(2\pi - \frac{\pi}{3}\right)$
$\sec \left(2\pi - \frac{\pi}{6}\right)$
$\cot \left(2\pi - \frac{3\pi}{4}\right)$

Problem set

Evaluate the following

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

Problem set

Evaluate the following

$\sin \left(122\pi\right)$
$\sec \left(29\pi\right)$
$\csc \left(1000\pi+\frac{\pi}{2}\right)$
$\sin \left(77\pi+\frac{\pi}{4}\right)$
$\tan \left(101157\pi+\frac{\pi}{6}\right)$
$\tan \left(101157\pi-\frac{\pi}{6}\right)$

Problem set

Evaluate the following

$\csc \left(\frac{14\pi}{3}\right)$
$\tan \left(\frac{19\pi}{4}\right)$
$\sec \left(\frac{31\pi}{6}\right)$
$\cot \left(\frac{62\pi}{3}\right)$

Problem set

Evaluate the following

$\tan (960^\circ)$
$\sec (1380^\circ)$
$\csc (945^\circ)$
$\cot (870^\circ)$
$\sec (900^\circ)$

Negative angles

Problem set

Evaluate the following

$\sin (-30^\circ)$
$\cos (-60^\circ)$
$\tan (-135^\circ)$
$\cos (-315^\circ)$
$\sec (-495^\circ)$
$\csc (-675^\circ)$
$\cot (-1500^\circ)$
$\sec (-1290^\circ)$
$\csc (-810^\circ)$

Problem set

Evaluate the following

$\cot \left(-\frac{\pi}{4}\right)$
$\csc \left(-\frac{\pi}{6}\right)$
$\sec \left(-\frac{\pi}{3}\right)$
$\csc \left(-\frac{\pi}{2}\right)$
$\sec \left(-2\pi\right)$
$\cot \left(-\pi\right)$

Problem set

Evaluate the following

$\csc \left(-\frac{2\pi}{3}\right)$
$\cos \left(-\frac{5\pi}{4}\right)$
$\sec \left(-\frac{5\pi}{6}\right)$

Problem set

$-\frac{7\pi}{3}$
$-\frac{13\pi}{6}$
$-\frac{7\pi}{4}$
$-\frac{11\pi}{6}$
$-\frac{8\pi}{3}$

Problem set

$-\frac{11\pi}{3}$
$-\frac{83\pi}{4}$
$-\frac{62\pi}{3}$
$-\frac{37\pi}{6}$
$-\frac{45\pi}{4}$

Problem set

$-\frac{25\pi}{6}$
$-\frac{140\pi}{60}$
$-\frac{17\pi}{6}$
$-\frac{14\pi}{3}$

Problem set

Evaluate the following

$\tan \left(-50\pi\right)$
$\sec \left(-1001\pi\right)$
$\cot \left(-1001\pi+\frac{\pi}{2}\right)$
$\sec \left(-8123\pi+\frac{\pi}{6}\right)$
$\tan \left(-77\pi+\frac{\pi}{3}\right)$

Problem set

Evaluate the following

$\tan \left(-2324\pi+\frac{\pi}{4}\right)$
$\cot \left(-4127\pi-\frac{\pi}{4}\right)$
$\sec \left(-1078\pi-\frac{\pi}{3}\right)$
$\csc \left(-3333\pi+\frac{\pi}{3}\right)$

Problem set

Evaluate the following

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

Miscellaneous

Exercises

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.

$\sin \theta = \cos \theta$
$\tan \theta = \cot \theta$
$2\tan \theta = \sec \theta$
$\sec \theta = \sqrt{3}\csc \theta$

Problem set

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

Express $\tan\theta$ in terms of $\sin\theta$ and $\cos\theta$ .
Express $\cot\theta$ in terms of $\sin\theta$ and $\cos\theta$ .
Express $\sin\theta$ in terms of $\cot\theta$ and $\cos\theta$ .
Express $\cos\theta$ in terms of $\cot\theta$ and $\sin\theta$ .
Express $\cot\theta$ in terms of $\sec\theta$ and $\cos\left(90^\circ-\theta\right)$ .
Express $\sec\theta$ in terms of $\sin\theta$ and $\tan\left(90^\circ-\theta\right)$ .
Express $\csc\theta$ in terms of $\tan\theta$ and $\sin\left(90^\circ-\theta\right)$ .

Problem set

Find the reference angle of each of the following.

$210^\circ$
$-300^\circ$
$-360^\circ\times 2 - 45^\circ$
$180^\circ\times 5 - 30^\circ$
$-180^\circ\times 5 + 60^\circ$
$480^\circ$
$-855^\circ$
$-1110^\circ$

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.

$\cos (495^\circ)$
$\tan (960^\circ)$
$\csc (1380^\circ)$
$\sin (-480^\circ)$
$\sec (-960^\circ)$
$\cot (-1395^\circ)$

Problem set

Find the reference angle of each of the following.

$119\pi+\frac{\pi}{6}$
$119\pi-\frac{\pi}{6}$
$2198\pi-\frac{\pi}{3}$
$2198\pi+\frac{\pi}{3}$
$\frac{7\pi}{4}$
$-\frac{19\pi}{4}$
$\frac{31\pi}{6}$
$-\frac{14\pi}{3}$

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 $\cos\theta$ .

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

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.

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

Problem set

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

Problem set

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