# 3.2 Sum and difference identities  (Page 3/6)

 Page 3 / 6

## Finding the exact value of an expression involving an inverse trigonometric function

Find the exact value of $\text{\hspace{0.17em}}\mathrm{sin}\left({\mathrm{cos}}^{-1}\text{\hspace{0.17em}}\frac{1}{2}+{\mathrm{sin}}^{-1}\text{\hspace{0.17em}}\frac{3}{5}\right).$

The pattern displayed in this problem is $\text{\hspace{0.17em}}\mathrm{sin}\left(\alpha +\beta \right).\text{\hspace{0.17em}}$ Let $\text{\hspace{0.17em}}\alpha ={\mathrm{cos}}^{-1}\frac{1}{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\beta ={\mathrm{sin}}^{-1}\frac{3}{5}.\text{\hspace{0.17em}}$ Then we can write

$\begin{array}{l}\hfill \\ \mathrm{cos}\text{\hspace{0.17em}}\alpha =\frac{1}{2},0\le \alpha \le \pi \hfill \\ \mathrm{sin}\text{\hspace{0.17em}}\beta =\frac{3}{5},-\frac{\pi }{2}\le \beta \le \frac{\pi }{2}\hfill \end{array}$

We will use the Pythagorean identities to find $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta .$

Using the sum formula for sine,

## Using the sum and difference formulas for tangent

Finding exact values for the tangent of the sum or difference of two angles is a little more complicated, but again, it is a matter of recognizing the pattern.

Finding the sum of two angles formula for tangent involves taking quotient of the sum formulas for sine and cosine and simplifying. Recall, $\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}x=\frac{\mathrm{sin}\text{\hspace{0.17em}}x}{\mathrm{cos}\text{\hspace{0.17em}}x},\mathrm{cos}\text{\hspace{0.17em}}x\ne 0.$

Let’s derive the sum formula for tangent.

We can derive the difference formula for tangent in a similar way.

## Sum and difference formulas for tangent

The sum and difference formulas for tangent are:

$\mathrm{tan}\left(\alpha +\beta \right)=\frac{\mathrm{tan}\text{\hspace{0.17em}}\alpha +\mathrm{tan}\text{\hspace{0.17em}}\beta }{1-\mathrm{tan}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\beta }$
$\mathrm{tan}\left(\alpha -\beta \right)=\frac{\mathrm{tan}\text{\hspace{0.17em}}\alpha -\mathrm{tan}\text{\hspace{0.17em}}\beta }{1+\mathrm{tan}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\beta }$

Given two angles, find the tangent of the sum of the angles.

1. Write the sum formula for tangent.
2. Substitute the given angles into the formula.
3. Simplify.

## Finding the exact value of an expression involving tangent

Find the exact value of $\text{\hspace{0.17em}}\mathrm{tan}\left(\frac{\pi }{6}+\frac{\pi }{4}\right).$

Let’s first write the sum formula for tangent and substitute the given angles into the formula.

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{tan}\left(\alpha +\beta \right)=\frac{\mathrm{tan}\text{\hspace{0.17em}}\alpha +\mathrm{tan}\text{\hspace{0.17em}}\beta }{1-\mathrm{tan}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\beta }\hfill \\ \mathrm{tan}\left(\frac{\pi }{6}+\frac{\pi }{4}\right)=\frac{\mathrm{tan}\left(\frac{\pi }{6}\right)+\mathrm{tan}\left(\frac{\pi }{4}\right)}{1-\left(\mathrm{tan}\left(\frac{\pi }{6}\right)\right)\left(\mathrm{tan}\left(\frac{\pi }{4}\right)\right)}\hfill \end{array}$

Next, we determine the individual tangents within the formula:

$\mathrm{tan}\left(\frac{\pi }{6}\right)=\frac{1}{\sqrt{3}},\mathrm{tan}\left(\frac{\pi }{4}\right)=1$

So we have

Find the exact value of $\text{\hspace{0.17em}}\mathrm{tan}\left(\frac{2\pi }{3}+\frac{\pi }{4}\right).$

$\frac{1-\sqrt{3}}{1+\sqrt{3}}$

## Finding multiple sums and differences of angles

Given find

1. $\mathrm{sin}\left(\alpha +\beta \right)$
2. $\mathrm{cos}\left(\alpha +\beta \right)$
3. $\mathrm{tan}\left(\alpha +\beta \right)$
4. $\mathrm{tan}\left(\alpha -\beta \right)$

We can use the sum and difference formulas to identify the sum or difference of angles when the ratio of sine, cosine, or tangent is provided for each of the individual angles. To do so, we construct what is called a reference triangle to help find each component of the sum and difference formulas.

1. To find $\text{\hspace{0.17em}}\mathrm{sin}\left(\alpha +\beta \right),$ we begin with $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\alpha =\frac{3}{5}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}0<\alpha <\frac{\pi }{2}.\text{\hspace{0.17em}}$ The side opposite $\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}$ has length 3, the hypotenuse has length 5, and $\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}$ is in the first quadrant. See [link] . Using the Pythagorean Theorem, we can find the length of side $\text{\hspace{0.17em}}a:$

Since $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta =-\frac{5}{13}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\pi <\beta <\frac{3\pi }{2},$ the side adjacent to $\text{\hspace{0.17em}}\beta \text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}-5,$ the hypotenuse is 13, and $\text{\hspace{0.17em}}\beta \text{\hspace{0.17em}}$ is in the third quadrant. See [link] . Again, using the Pythagorean Theorem, we have

Since $\text{\hspace{0.17em}}\beta \text{\hspace{0.17em}}$ is in the third quadrant, $\text{\hspace{0.17em}}a=–12.$

The next step is finding the cosine of $\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}$ and the sine of $\text{\hspace{0.17em}}\beta .\text{\hspace{0.17em}}$ The cosine of $\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}$ is the adjacent side over the hypotenuse. We can find it from the triangle in [link] : $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\alpha =\frac{4}{5}.\text{\hspace{0.17em}}$ We can also find the sine of $\text{\hspace{0.17em}}\beta \text{\hspace{0.17em}}$ from the triangle in [link] , as opposite side over the hypotenuse: $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\beta =-\frac{12}{13}.\text{\hspace{0.17em}}$ Now we are ready to evaluate $\text{\hspace{0.17em}}\mathrm{sin}\left(\alpha +\beta \right).$

2. We can find $\text{\hspace{0.17em}}\mathrm{cos}\left(\alpha +\beta \right)\text{\hspace{0.17em}}$ in a similar manner. We substitute the values according to the formula.
3. For $\text{\hspace{0.17em}}\mathrm{tan}\left(\alpha +\beta \right),$ if $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\alpha =\frac{3}{5}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\alpha =\frac{4}{5},$ then
$\mathrm{tan}\text{\hspace{0.17em}}\alpha =\frac{\frac{3}{5}}{\frac{4}{5}}=\frac{3}{4}$

If $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\beta =-\frac{12}{13}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta =-\frac{5}{13},$ then

$\mathrm{tan}\text{\hspace{0.17em}}\beta =\frac{\frac{-12}{13}}{\frac{-5}{13}}=\frac{12}{5}$

Then,

4. To find $\text{\hspace{0.17em}}\mathrm{tan}\left(\alpha -\beta \right),$ we have the values we need. We can substitute them in and evaluate.

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!

#### Get Jobilize Job Search Mobile App in your pocket Now! By By OpenStax By Michael Nelson By John Gabrieli By Brooke Delaney By Yacoub Jayoghli By OpenStax By Brooke Delaney By Marion Cabalfin By OpenStax