# Trigonometric Identities

When students are first exposed to trigonometric identities, they are often given a list of formulas, which they are asked to memorize. Here is a way for you to remember many of these ideas.

Start by drawing a regular hexagon and connect each of the vertices to the center. Label the left-most vertex $\tan$, label the bottom left vertex $\sin$, and label the bottom right vertex $\cos$. Label the center 1. Now, to figure out how to label the remaining vertices, simply look at the diagonally opposite vertex and label it as the reciprocal. You should get the following diagram: Hexagon of Trigonometric Functions

(Note: $\frac {1}{ \cos \theta} = \csc \theta$, which is sometimes denoted as cosec $\theta$. Either version is valid, and we will be using $\csc$ in this post.)

Now, let's figure out some properties of this hexagon.

Property 1: How do the labels on the endpoints of a diameter relate? Recall that to establish the labels, we labelled the diagonally opposite vertex as the reciprocal. Hence, the product of the labels on a diameter is 1, which corresponds to the center vertex.

Property 2: How do we relate vertices that are connected by edges? Consider the central vertex 1. Moving directly to the left is equivalent to multiplying by $\tan$, moving to the lower right is equivalent to multiplying by $\cos$, moving to the lower left is equivalent to multiplying by $\sin$. This seems obvious when we're at vertex 1, and in fact holds true for any other vertex. For example, moving left from $\cos$ brings us to $\sin$, which corresponds to $\cos \theta \times \tan \theta = \sin \theta$. There are similar statements for moving to the right, upper left and upper right.

Property 3: What property do we get when reading off the 3 vertices in an anti-clockwise manner? With our 3 starting vertices, we know that $\tan \theta = \frac {\sin \theta} { \cos \theta}$. This behavior holds true for the rest, that the first vertex is equal to the second divided by the third. For example, another set of 3 vertices is $\cos, \cot, \csc$, and you should be able to verify that $\cos \theta = \frac { \cot \theta} { \csc \theta}$.

Property 4: Recall the Pythagorean identity which states that $\sin^2 \theta + \cos ^2 \theta = 1^2$. How is this expressed in the hexagon? If we look at the bottom upright triangle, we see it it has vertices of $\sin, \cos$ at the base, and $1$ at the top. In fact, given any other upright triangle, a similar relation holds. For example, with the right upright triangle, we get $1^2 + \cot ^2 \theta = \csc ^2 \theta$, and with the left upright triangle, we get $\tan^2 \theta + 1 = \sec^2 \theta$.

## Worked Examples

### 1. What is $\frac {\sec \theta} { \tan \theta}$?

Solution: By property 3, we know that $\csc \theta = \frac { \sec \theta} { \tan \theta}$.

### 2. Why does property 4 hold?

Solution: We already know that $\sin ^2 \theta + \cos^2 \theta = 1^2$. Consider what happens when we shift this triangle to the upper right. We are simply multiplying each vertex by $\sec \theta$. Hence, the equivalent identity is that $\sec^2 \theta [ \sin ^2 \theta + \cos^2 \theta] = \sec^2 \theta \cdot 1^2$, which becomes $\tan^2 \theta + 1^2 = \sec^2 \theta$.

### 3. What other properties are there?

Solution: This is an open ended question. There are as many properties as you can find for yourself. We will state one more, which is related to Property 3 (and actually is simply a different way of expressing the same idea):

Property 5: If we read a set of 3 vertices off in a clockwise-order, we get that the first vertex is equal to the second over the third. For example, $\sin, \tan , \sec$ are 3 consecutive vertices in clockwise order, so this property gives us that $\sin \theta = \frac { \tan \theta} { \sec \theta}$. Note by Arron Kau
7 years, 3 months ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
• Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$

## Comments

Sort by:

Top Newest

Where's the "Test Yourself" section?

- 7 years, 1 month ago

Log in to reply

It's not present anymore, but you can always get lots of Trig Identity problems here: https://brilliant.org/tags/trigonometricidentities/.

Staff - 7 years, 1 month ago

Log in to reply

where is the challenge a friend option????

- 6 years, 8 months ago

Log in to reply

Thanks a lot. A nice way to remember.

- 7 years ago

Log in to reply

Brilliant

- 6 years, 6 months ago

Log in to reply

this is really GOOD!!!Thanks but could you just explain more about properties of property 4?

- 6 years, 4 months ago

Log in to reply

Prove that 1+cot^2 theta/1+cosec theta=1/sin theta

- 6 years, 1 month ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...