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### Course: Precalculus > Unit 2

Lesson 10: Using trigonometric identities- Finding trig values using angle addition identities
- Using the tangent angle addition identity
- Find trig values using angle addition identities
- Using trig angle addition identities: finding side lengths
- Using trig angle addition identities: manipulating expressions
- Using trigonometric identities
- Trig identity reference
- Trigonometry: FAQ

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# Using the tangent angle addition identity

Find the tangent of 13pi/12 without a calculator using the tangent angle addition identity. Created by Sal Khan.

## Want to join the conversation?

- why are the practice problems so much harder than what we learn in the video?

i feel like there are not enough videos here to go ahead to the parctice problem because he does not show how to solve so many of them(10 votes)- You could read the article at the end of this section, this covers equations ranging from the
*inverse*of sin to the*reciprocal*of tangent and the equations to solve the problems.(5 votes)

- It should be noted that it's possible to simplify the answer even further. Sal came up with (sqrt(3)-1)/(sqrt(3)+1), which is not wrong; however, if you multiply both numerator and denominator by (sqrt(3)-1), the numerator becomes (sqrt(3)-1)^2, which simplifies to 4-2sqrt(3), while the denominator simplifies to the difference of two squares, 3-1 = 2. Thus the entire expression can be simplified to 2-sqrt(3).(10 votes)
- whoa whoa whoa, where the heck did he get 15pi/12 - 2pi/12? is he just picking randomn fractions(angles or whatver)? By that reasoning, why can't I just use 3pi/12 plus 20pi/12?(6 votes)
- He picked them because 15pi/12-2pi/12=13pi/12, which is the angle we are trying to evaluate. Why did he pick those two angles to find the difference of (when there are many others that would also equal 13pi/12)? Because of what they simplify down to. The 15pi/12 simplifies to 5pi/4. We know how to find tan(5pi/4), which is 1. The same thing goes for 2pi/12. It simplifies to pi/6, and tan(pi/6)=1/sqrt(3).

Hope this helps!(4 votes)

- Isn't the slope change in y over change in x? Why does he use one point and just does y over x = slope?(5 votes)
- The line goes through the origin (0,0) and the point (x,y). So, the complete formula would be (y-0)/(x-0)=y/x.

So, if you are finding the slope of a line that goes through the origin, all you have to do is take y/x.

Hope this helps!(3 votes)

- When Sal says the slope of the tangent on the unit circle is just the radius for 5pi/4 = 1, why is it different for pi/6?(4 votes)
- Since tangent is opposite divided by adjacent, the values in the 45-45-90 triangle are both the same so it'll just be 1.(4 votes)

- Why did Sal rationalize the variables last? I mean, I can see why that was a good idea since it was faster than just rationalizing them from the start, but is there a rule on when to rationalize or is it just common sense on when it's a good idea to rationalize now than later?

Edit: I notice he didn't have to rationalize this way.(4 votes)- It's always easier to work with smaller numbers, so personally I lean to rationalizing denominators afterwards unless I see some obvious benefit of doing it beforehand.(4 votes)

- In1:16, Sal said we've proven tangent identities in another video. Where can I find that video?

I was trying to search the tangent identities but couldn't find a video with the sum and difference identities of the tangent.(5 votes) - How would you solve an equation such as (cos(x))/(1+csc(x))*(1-csc(x))/(1-csc(x))?

I end up always getting these wrong, and they always answer (such as with this one) with something like tan(x)-tan(x)sin(x). why and how did they get to that answer??(3 votes) - Find cos(13pi/12) exactly using an angle addition or subtraction formla.

I broke up (13pi/12) to (4pi/12 + 9pi/12). This futher simplifies to (pi/3 + 3pi/4). pi/3 is angle 60 degrees in the first quadrant and 3pi/4 is angle 45 degrees in the 2nd quadrant.

I used the addition formula cos(x+y) = cos(x)cos(y)-sin(x)sin(y). Assigned x to pi/3 and y to 3pi/4.

Cos(pi/3) = cos(60) = adjacent/hypotenuse = +x/1 = .5/1 = .5 (1st quadrant)

Cos(3pi/4) = cos(45) = adjacent/hypotenuse = +y/1 = sqroot2/2 divided by 1 = sqroot2/2 (2nd quadrant)

Sin(pi/3) = sin(60) = opposite/hypotenuse = +y/1 = sqroot3/2 divided by 1 = sqroot3/2 (1st quadrant)

Sin(3pi/4) = sin(45) = opposite / hypotenuse = -x/1 =

-sqroot2/2 divide by 1 = -sqroot2/2 (2nd quadrant)

Appling the sin and cosine values to the formula cos(x+Y) i got the answer sqroot2/4 + sqroot6/4.

The correct answer to the problem given is -sqroot2/4 -sqroot6/4.

Where did I make my mistake? Please help.(2 votes)- So there two definitions of the trignometric functions one based on SOHCAHTOA and the other the definitions of the unit circle.

For this question there is no need to use SOHCAHTOA so I would avoid it.

There are two mistakes.

1) cos(3pi/4) = sqrt(2)/2 should be cos(3pi/4) = -sqrt(2)/2

2) sin(3pi/4) = -sqrt(2)/2 should be sin(3pi/4) = sqrt(2)/2

This will make sense when you think about it. By definition cos(theta) = x and sin(theta) = y. But in the second quadrant sin(theta) is positive and cos(theta) is negative.

Fixing that you should arrive at the correct answer.

As a side note I would avoid writing sqroot instead write sqrt as it is more commonly used. Also use parenthesis when using sqrt as it not clear from your post what argument that is the input to the sqrt function is meant to be.(2 votes)

- I did it differently and got to the same answer faster using 3pi/4 + pi/3(1 vote)

## Video transcript

- [Instructor] In this video,
we're going to try to compute what tangent of 13 pi over 12
is without using a calculator. But I will give you a few hints. First of all, you can rewrite
tangent of 13 pi over 12 as tangent of, instead of 13 pi over
12, we can express that in terms of angles where we
might be able to figure out the tangents just based
on other things we know about the unit circle. 13 pi over 12 is the same
thing as 15 pi over 12 minus two pi over 12, which is the same thing as the tangent of five pi over four minus pi over six, or we could even view it as
plus negative pi over six. So that's my hint right over there. So pause this video and
see if you can keep going with this train of reasoning to evaluate what tangent
of 13 pi over 12 is without using a calculator. All right, now let's
keep on going together. Well, we already know what the tangent of the sum of two angles are. We've proven that in another video. We know that this is going to be equal to the tangent of the
first of these angles, five pi over four, plus the tangent of the second angle, tangent of negative pi over six, all of that is going to be over one minus tangent of the first angle, five pi over four, times the tangent of the second angle, negative pi over six. And so now we can break
out our unit circles to figure out what these things are. So I have pre-put some unit circles here. And so let's first think about what five pi over four looks like. Pi over four, you might already associate
it with 45 degrees. That's pi over four right over there. Two pi over four would get you here. Three pi over fours would get you there. Four pi over four, which
the same thing as pi, gets us over there. Five pi over four would
get us right about there. Now, you might already recognize
the tangent of an angle as the slope of the radius, and so you might already be able to intuit that the tangent here is going to be one, but we can also break out
our knowledge of triangles in the unit circle to figure this out if you didn't realize that. So what we need to do is
figure out the coordinates of that point right over there, and to help us do that, we can set up a little
bit of a right triangle, which you might immediately recognize is a 45-45-90 triangle. How do I know that? Well five pi over four, remember we go four pi
over four to get here, then we have one more pi
over four to go down here. So this angle right over
here is pi over four or you could view it as 45 degrees. And of course, if that's
45 degrees and that's 90, then this has to be 45 degrees 'cause they all add up to 180 degrees. And we know a triangle like
this by the Pythagorean theorem, if our hypothesis is one, each of the other two sides is square root of two over
two times the hypotenuse. So this is square root of two over two and then this is square
root of two over two. Now, if we think about the coordinates, our X coordinate is square
root of two over two in the negative direction. So our X coordinate is negative
square root of two over two and our Y coordinate is square root of two over two going down, so that's also negative
square root of two over two. And the tangent is just the Y coordinate over
the X coordinate here. So the tangent is just going to be negative square root of two over two over negative square root of two over two, which is once again, one,
which was our intuition. So we can write the tangent
of five pi over four is equal to one. And then what about negative pi over six? Well, negative pi over six, you might recognize pi over
six as being a 30 degree angle. Pi is 180 degrees, so
divided by six is 30 degrees, and negative pi over six would be going 30 degrees
below the positive x-axis. So it would look just like that. And as we said, this angle
right over here is pi over six, which you can also view
as a 30 degree angle. If we were to drop a
perpendicular right over here, you might immediately recognize this as a 30-60-90 triangle. We know if the hypotenuse
is of length one, the side opposite the 30 degree side is 1/2 the hypotenuse, and then the longer non-hypotenuse side is going to be square root of
three times the shorter side, so square root of three over two. And so our coordinate's right over here, we are moving square
root of three over two in the positive X direction, square root of three over two, and then we're going negative
1/2 in the Y direction, so we put negative 1/2 right over there. And so now we know that the
tangent of negative pi over six is going to be equal to negative 1/2 over square root of three over two, which is the same thing
as negative 1/2 times, let me write it this way, negative 1/2 times two over the square root of three, which is equal to negative one over the square root of three. This right over here is one. We saw that there. This right over here is one. And then this right over here is negative one over the
square root of three. And then this is negative one
over the square root of three. And so I can rewrite this
entire expression as being equal to one minus one over the square to three, all of that over one, and then I have a negative
here and a negative here, so then that becomes, a
negative times a negative, so positive one plus one over
the square root of three. And if we multiply both the
numerator and the denominator by square root of three, what we're going to get in the numerator is square root of three minus one and then our denominator is going to be square root of three plus one. And we are done. That's tangent of 13 pi over
12 without using a calculator.