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### Course: Precalculus > Unit 2

Lesson 9: Angle addition identities- Trig angle addition identities
- Using the cosine angle addition identity
- Using the cosine double-angle identity
- Using the trig angle addition identities
- Proof of the sine angle addition identity
- Proof of the cosine angle addition identity
- Proof of the tangent angle sum and difference identities

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# Trig angle addition identities

Sal reviews 6 related trigonometric angle addition identities: sin(a+b), sin(a-c), cos(a+b), cos(a-b), cos(2a), and sin(2a). Created by Sal Khan.

## Want to join the conversation?

- Sal goes over a whole lot of identities in this video - which of these, if any, are you generally expected to memorize in, say, a high school trigonometry course? How regularly are they used in the rest of pre-calculus and calculus?(92 votes)
- Ideally speaking you would have reference sheet you could use. There is always the risk a memory pathway in your brain could change without you realising. Pretty much all the identities are important in some way.

A list of identities are given by:

https://www.khanacademy.org/math/trigonometry/trig-equations-and-identities/using-trig-identities/a/trig-identity-reference**Unit Circle Definitions**:

To add to the article, the unit circle definitions of the trigonometric function are important namely:

sin(theta) = y

cos(theta) = x

tan(theta) = y/x

There are some other memory tricks you can use.**Complementary Angle Identities**:

co is a prefix the stands for complementary. For example in cosine i.e. co-sine.

Therefore sin(theta) = cos(90-theta).

Similarly sin(90-theta) = cos(theta)

cot(90-theta) = tan(theta)

tan(90-theta) = cot(theta)

cosec(90-theta) = sec(theta)

sec(90-theta) = cosec(theta)

The complementary angle identities can be proven by proving they hold for each quadrant than making use of the period to prove they hold for all real numbers.

**Pythagorean Identities**:

The pythagorean identities come from equation of the circle

x^2 + y^2 = 1

Using the unit circle definition (see above) you will end up with cos^2(theta)+sin^2(theta)=1 from direct substitution.

A man with a tan is sexy (sec C)

1 + tan^2(theta) = sec^2(theta)

A man in a cot is cosy (cosec)

1+ cot^2(theta) = cosec^2(theta)**Reciprocal**:

For reciprocal identities observe the third letter of cosecant, cotangent, secant.

cosec(theta) = 1/sin(theta) # third letter of cosecant is s

sec(theta) = 1/cos(theta) # third letter of secant is c

cot(theta) = 1/tan(theta) # third letter of cotangent is t**Trig Ratio Identities**:

tan(theta) = sin(theta)/cos(theta)

cot(theta) = cos(theta)/sin(theta)

You will need to know the definition of period, odd/even functions. I recommend knowing what sin, cos, tan look like. This will help you know whether function is odd or even. For the period you should be able to derive it from the unit circle definitions using congruency testing. (period of sin and cos is 2pi. period of tan is pi)**Identities for integration**:

The two identities below are important for integration in calculus.

For cos, cos is positive

cos^2(theta) = [1+cos(2*theta)]/2

For sine, cos is negative

sin^2(theta) = [1-cos(2*theta)]/2

For the 6 identities mention in this video with knowledge of even/odd functions and practice you will get comfortable with them eventually, however I recommend using reference sheet if possible as mentioned earlier.(9 votes)

- why cos(-x)=cos(x)(46 votes)
- It may be helpful to think about the graphs of sine and cosine. At x = 0, sine starts at 0 and goes up to 1, while cosine starts at 1 and goes down. Think about what happens on the negative side though. On the sine graph, for negative x's you are getting negative y's, as it passes below the x-axis, while the cosine keeps giving you positive y's as it starts down toward the x-axis.

Sine graph: http://www.wolframalpha.com/share/clip?f=d41d8cd98f00b204e9800998ecf8427eelg3k45l05

Cosine graph: http://www.wolframalpha.com/share/clip?f=d41d8cd98f00b204e9800998ecf8427esflsi0ko9p(21 votes)

- How do you verify trigonometric identities? Like cos x sec x = 1 or 1-tan^2x = 2-sec^2x(47 votes)
- 1) sec x = 1/(cos x), so the first equation simplifies to (cos x)/ (cos x), which equals 1.

2) 1 - tan^2 x = 1 - (sin^2 x)/(cos^2 x). Making common denominators and making one fraction => (cos^2 x - sin ^2 x)/(cos^2 x). Since we know cos^2 x + sin^2 x = 1, then sin^2 x = 1 - cos^2 x. Substition => (cos^2 x - (1 - cos^2 x))/(cos^2 x) = (2cos^2 x - 1)/(cos^2 x).

On the right side of original equation, 2 - sec^2 x = 2 - (1/(cos^2 x)). Making common denominators and one fraction => (2cos^2 x -1)/(cos^2 x).

Now we have both sides the same, therefore the identity is proven.(63 votes)

- A0:21, Sal says that he is assuming we already know a bunch of properties, like the fact that the sin(a+b) = sin a + cos b. I don't recall seeing this shown or proved anywhere earlier in the Trig material. Did I just forget it? Or has this not been shown yet? A few times, I've seen the videos ordered such that a video that assumes we already know something is placed before the video that explains it. That gets confusing. ;u;

If I did just forget the video where this is shown earlier, can someone point me to the videos I should review? I got a bit lost watching this 'cause I was wondering about the properties we are supposed to know already!(55 votes)- Hi Christopher,

Go here : https://www.khanacademy.org/math/algebra-home/precalculus/trig-equations-and-identities-precalc

And under the group: Introduction to the trigonometric angle addition identities

There should be videos for what you are looking for.(15 votes)

- Where are the lessons that cover the things we are assumed to know in this video?

Is there a problem with the order these are presented?

We are starting an 'INTRO to trig angle addition identities

with a 'REVIEW of trig angle addition identities' we are supposed to know.

I have been working systematically thru this material for months and didn't do this yet.

Am I missing something?!

Thanks for any help getting orientated.(22 votes)- As of posting this comment, the videos are a bit out order. Of the 3 identities you are assumed to know already, only sin(a + b) and cos(a + b) haven't previously been taught in the precacl course.

The videos (which should be before this one) is later in this lesson

https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:trig/x9e81a4f98389efdf:angle-addition/v/proof-angle-addition-sine

https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:trig/x9e81a4f98389efdf:angle-addition/v/proof-angle-addition-cosine(27 votes)

- Where are hyperbolic functions such as hyperbolic sine, hyperbolic cosine and hyperbolic tangent used for? I would really appreciate if someone could give an example with an explanation along with it :-) I'm kinda confused because we never discussed the hyperbolic functions in trigonometry class. I'm not even sure if they are related to trigonometry. Thanks in advance!(20 votes)
- The hyperbolic functions pop up in trigonometry when you start defining the trig functions for complex arguments. For example, sin z = sinh(i*z) / i = - i*sinh(i*z).

Fun fact: any chain or string that is attached from its endpoints forms a catenary, not a parabola. A catenary is of the form y = a * cosh(x/a).

http://en.wikipedia.org/wiki/Catenary(14 votes)

- What does Sal mean, when he says that cosine is an even function and sine is an odd function?(2 votes)
- He means that cos(-x) = cos(x) so it is even.

In the same way that x^2 is an even function, cos(x) behaves similarly.

If you take a negative number like -2 and square it the result is always positive

So if you take cos(-1) it will get you the same result as taking the cos(1)

However, with the sin function it is different.

-sin(x) = sin(-x)

Which basically means that sin(1) and sin(-1) will produce two different answers.

Someone on here could most probably explain it better than I did.(25 votes)

- In what video were these identities discussed? I have been following the Trigonometry course since the beginning and don't remember these identities being taught.

Thanks!(13 votes)- So just accept the theorems presented in the video at the beginning.

Proof of the trig angle addition will be given later on as well as exercises. To put it another way you think of the video you are watching as the introduction.(0 votes)

- are we expected to memorize all of these?(7 votes)
- Unfortunately, yes.

You can remember the addition identity for sine as this phrase: “SUMthing that switches.” The phrase reminds you that you have to**swap**the sin and cos and**add**.

And for cosine, it is the opposite: you find the**difference**between taking the cos of**both**and the sin of**both**.(10 votes)

- Where can I find, on this website, the previous videos that sal mentioned on the topic of identities?(11 votes)

## Video transcript

i've already made a handful
of videos that covers what I'm going to cover, the
trigonometric identities I'm going to cover in this video. The reason why I'm doing it is
that I'm in need of review myself because I was doing some
calculus problems that required me to know this, and I have
better recording software now so I thought two birds with one
stone, let me rerecord a video and kind of refresh
things in my own mind. So the trig identities that I'm
going to assume that we know because I've already made
videos on them and they're a little bit involved to remember
or to prove, are that the sine of a plus b is equal to the
sine of a times the cosine of b plus the sine of b
times the cosine of a. That's the first one, I assume,
going into this video we know. And then if we wanted to know
the sine of-- well, I'll just write it a little differently. What if I wanted to figure out
the sine of a plus-- I'll write it this way-- minus c? Which is the same thing
as a minus c, right? Well, we could just use this
formula up here to say well, that's equal to the sine of a
times the cosine of minus c plus the sine of minus c
times the cosine of a. And we know, and I guess this
is another assumption that we're going to have to have
going into this video, that the cosine of minus c is equal
to just the cosine of c. That the cosine is
an even function. And you could look at that by
looking at the graph of the cosine function, or even at
the unit circle itself. And that the sine is
an odd function. That the sine of minus
c is actually equal to minus sine of c. So we can use both of that
information to rewrite the second line up here to say that
the sine of a minus c is equal to the sine of a times
the cosine of c. Because cosine of minus
c is the same thing as the cosine of c. Times the cosine of c. And then, minus the sine of c. Instead of writing this,
I could write this. Minus the sine of c
times the cosine of a. So that we kind of pseudo
proved this by knowing this and this ahead of time. Fair enough. And I'm going to use all of
these to kind of prove a bunch of more trig identities
that I'm going to need. So the other trig identity is
that the cosine of a plus b is equal to the cosine of a-- you
don't mix up the cosines and the sines in this situation. Cosine of a times
the sine of b. And this is minus--
well, sorry. I just said you don't mix it
up and then I mixed them up. Times the cosine of b minus
sine of a times the sine of b. Now, if you wanted to know what
the cosine of a minus b is, well, you use these
same properties. Cosine of minus b, that's still
going to be cosine on b. So that's going to be the
cosine of a times the cosine-- cosine of minus b is the
same thing as cosine of b. But here you're going to have
sine of minus b, which is the same thing as the
minus sine of b. And that minus will cancel that
out, so it'll be plus sine of a times the sine of b. So it's a little tricky. When you have a plus sign
here you get a minus there. When you don't minus
sign there, you get a plus sign there. But fair enough. I don't want to dwell on that
too much because we have many more identities to show. So what if I wanted an
identity for let's say, the cosine of 2a? So the cosine of 2a. Well that's just the same thing
as the cosine of a plus a. And then we could use this
formula right up here. If my second a is just my b,
then this is just equal to cosine of a times the cosine
of a minus the sine of a times the sine of a. My b is also an a in this
situation, which I could rewrite as, this is equal to
the cosine squared of a. I just wrote cosine of a times
itself twice or times itself. Minus sine squared of a. This is one I guess
identity already. Cosine of 2a is equal to the
cosine squared of a minus the sine squared of a. Let me box off my identities
that we're showing in this video. So I just showed you that one. What if I'm not satisfied? What if I just want it
in terms of cosines? Well, we could break out
the unit circle definition of our trig functions. This is kind of the most
fundamental identity. The sine squared of a
plus the cosine squared of a is equal to 1. Or you could write that--
let me think of the best way to do this. You could write that the sine
squared of a is equal to 1 minus the cosine
sign squared of a. And then we could take this
and substitute it right here. So we could rewrite this
identity as being equal to the cosine squared of a minus
the sine squared of a. But the sine squared of
a is this right there. So minus-- I'll do it
in a different color. Minus 1 minus cosine
squared of a. That's what I just substituted
for the sine squared of a. And so this is equal to the
cosine squared of a minus 1 plus the cosine squared of a. Which is equal to--
we're just adding. I'll just continue
on the right. We have 1 cosine squared of a
plus another cosine squared of a, so it's 2 cosine
squared of a minus 1. And all of that is
equal to cosine of 2a. Now what if I wanted to get
an identity that gave me what cosine squared of
a is in terms of this? Well we could just
solve for that. If we add 1 to both sides of
this equation, actually, let me write this. This is one of our
other identities. But if we add 1 to both sides
of that equation we get 2 times the cosine squared of a is
equal to cosine of 2a plus 1. And if we divide both sides of
this by 2 we get the cosine squared of a is equal to 1/2--
now we could rearrange these just to do it-- times 1
plus the cosine of 2a. And we're done. And we have another identity. Cosine squared of a, sometimes
it's called the power reduction identity right there. Now what if we wanted
something in terms of the sine squared of a? Well then maybe we could go
back up here and we know from this identity that the sine
squared of a is equal to 1 minus cosine squared of a. Or we could have
gone the other way. We could have subtracted sine
squared of a from both sides and we could have gotten--
let me go down there. If I subtracted sine squared of
a from both sides you could get cosine squared of a is equal
to 1 minus sine squared of a. And then we could go back into
this formula right up here and we could write down-- and I'll
do it in this blue color. We could write down that the
cosine of 2a is equal to-- instead of writing a cosine
squared of a, I'll write this- is equal to 1 minus sine
squared of a minus sine squared of a. So my cosine of 2a is equal to? Let's see. I have a minus sine squared
of a minus another sine squared of a. So I have 1 minus 2
sine squared of a. So here's another identity. Another way to write
my cosine of 2a. We're discovering a lot of ways
to write our cosine of 2a. Now if we wanted to solve for
sine squared of 2a we could add it to both sides
of the equation. So let me do that and I'll
just write it here for the sake of saving space. Let me scroll down
a little bit. So I'm going to go here. If I just add 2 sine squared
of a to both sides of this, I get 2 sine squared of a plus
cosine of 2a is equal to 1. Subtract cosine of
2a from both sides. You get 2 sine squared of a is
equal to 1 minus cosine of 2a. Then you divide both sides of
this by 2 and you get sine squared of a is equal to 1/2
times 1 minus cosine of 2a. And we have our other discovery
I guess we could call it. Our finding. And it's interesting. It's always interesting
to look at the symmetry. Cosine squared-- they're
identical except for you have a plus 2a here for the cosine
squared and you have a minus cosine of 2a here for
the sine squared. So we've already found a
lot of interesting things. Let's see if we can do
anything with the sine of 2a. Let me pick a new color
here that I haven't used. Well, I've pretty much
used all my colors. So if I want to figure out the
sine of 2a, this is equal to the sine of a plus a. Which is equal to the sine of a
times the co-- well, I don't want to make it that thick. Times the cosine of a plus--
and this cosine of a, that's the second a. Actually, you could
view it that way. Plus the sine-- I'm just
using the sine of a plus b. Plus the sine of the
second a times the cosine of the first a. I just wrote the same thing
twice, so this is just people to 2 sine of a, cosine of a. That was a little bit easier. So sine of 2a is equal to that. So that's another result. I know I'm a little bit tired
by playing with all of these sine and cosines. And I was able to get all the
results that I needed for my calculus problem, so hopefully
this was a good review for you because it was a
good review for me. You can write these
things down. You can memorize them if you
want, but the really important take away is to realize that
you really can derive all of these formulas really from
these initial formulas that we just had. And even these, I also have
proofs to show you how to get these from just the basic
definitions of your trig functions.