Waste less time on Facebook — follow Brilliant.
×

Trigonometry: Basic Trig Identities

This post is part of a series of posts on Trigonometry. To see all the posts, click on the tag #TrigonometryTutorials below. This is the post you should read before you read this.

Here are a few trig identities that are very important to remember:

Identity 1: \[\sin^2 x+ \cos^2 x = 1\]

Proof: Use the Definition of the trig functions: \(\sin x = \frac oh\) and \(\cos \frac ah\)

\[\left(\frac oh\right)^2+\left(\frac ah\right)^2= 1\] \[\frac {o^2}{h^2} + \frac {a^2}{h^2} = 1\]

Multiply both sides by \(h^2\)

\[a^2+o^2=h^2\]

This is just the Pythagorean Theorem! So this holds true. \(\blacksquare\)

A variation of Identity 1 can be achieved by:

dividing both sides by \(\sin^2 x\)

\[1+ \cot^2 x = \csc x\]

dividing both sides by \(\cos^2 x\)

\[ \tan^2 x +1 = \sec x\]

Identity 2: \(\sin x = \cos (90^\circ-x)\)

In a Right angled triangle, if one of the angles is \(x\), The other angle will be \(90-x\) (the angle other than the right angle, that is.). The opposite angle of one angle is the adjacent of the other angle, therefore \(\sin x = \cos (90^\circ-x)\).

Using the same reasoning, these other identities can be obtained:

\[\cos x = \sin (90-x)\] \[\sec x = \csc (90-x)\] \[\csc x = \sec (90-x)\] \[\tan x = \cot(90-x)\] \[\cot x = \tan (90-x)\]

The Next post in this series is here

Note by Yan Yau Cheng
3 years, 11 months ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. 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 1

paragraph 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

There are no comments in this discussion.

×

Problem Loading...

Note Loading...

Set Loading...