Question 1

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?

Accepted Answer

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.

Question 2

A 0:21, 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!

Accepted Answer

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.

Question 3

How do you verify trigonometric identities? Like cos x sec x = 1 or 1-tan^2x = 2-sec^2x

Accepted Answer

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.

Question 4

why cos(-x)=cos(x)

Accepted Answer

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

Question 5

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.

Accepted Answer

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

Question 6

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!

Accepted Answer

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

Question 7

are we expected to memorize all of these?

Accepted Answer

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*.

Question 8

What does Sal mean, when he says that cosine is an even function and sine is an odd function?

Accepted Answer

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.

Course: Precalculus (Eureka Math/EngageNY) > Unit 4

Trig angle addition identities

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