Main content
Course: IIT JEE > Unit 1
Lesson 1: IIT JEE- Trig challenge problem: arithmetic progression
- IIT JEE perpendicular planes (part 1)
- IIT JEE perpendicular plane (part 2)
- IIT JEE complex root probability (part 1)
- IIT JEE complex root probability (part 2)
- IIT JEE position vectors
- IIT JEE integral limit
- IIT JEE algebraic manipulation
- IIT JEE function maxima
- IIT JEE diameter slope
- IIT JEE hairy trig and algebra (part 1)
- IIT JEE hairy trig and algebra (part 2)
- IIT JEE hairy trig and algebra (part 3)
- Challenging complex numbers problem (1 of 3)
- Challenging complex numbers problem (2 of 3)
- Challenging complex numbers problem (3 of 3)
- IIT JEE differentiability and boundedness
- IIT JEE integral with binomial expansion
- IIT JEE symmetric and skew-symmetric matrices
- IIT JEE trace and determinant
- IIT JEE divisible determinants
- Intersection of circle & hyperbola
- Common tangent of circle & hyperbola (1 of 5)
- Common tangent of circle & hyperbola (2 of 5)
- Common tangent of circle & hyperbola (3 of 5)
- Common tangent of circle & hyperbola (4 of 5)
- Common tangent of circle & hyperbola (5 of 5)
- Trig challenge problem: multiple constraints
- Trig challenge problem: maximum value
- Vector triple product expansion (very optional)
- IIT JEE lagrange's formula
- Representing a line tangent to a hyperbola
- 2010 IIT JEE Paper 1 Problem 50: Hyperbola eccentricity
- Normal vector from plane equation
- Point distance to plane
- Distance between planes
- Challenging complex numbers problem: complex determinant
- Series sum example
- Trig challenge problem: system of equations
- Simple differential equation example
© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice
IIT JEE hairy trig and algebra (part 1)
2010 Paper 1 problem 38 Hairy Trig and Algebra. Created by Sal Khan.
Want to join the conversation?
- The right way to do it is this :
cosC = (a^2+b^2-c^2)/2ab
Now subtract 1 from both the sides (some also like to call it componendo and dividendo)
cosC - 1 = (a^2+b^2 -c^2)/2ab -1
cosC -1 = (a^2+b^2 -c^2 -2ab)/2ab
Regroup
cosC -1 = [ (a^2+b^2 -2ab) -c^2]/2ab
cosC -1 = [ (a-b)^2 - c^2 ]/2ab
cosC -1 = [ (a-b+c)(a-b-c) ]/2ab
( since x^2 - y^2 = (x-y)(x+y) )
Now all you have do is do basic addition or subtractions :
a-b+c = 3x+3 = 3(x+1)
a-b-c = -(x-1)
so , cosC -1 = -3(x+1)(x-1)/2ab
now you know b = x^2-1 = (x-1)(x+1)
So (x-1)(x+1) cancels out from denominator and numerator.
cosC -1 = -3/2a
so 2a/-3 = 1/(cosC -1)
(by taking inverse on both the sides or you could say cross multiply)
so a= -3/2(cosC-1)
And put cosC = sqrt(3)/2
And tada ! you get a beautiful quadratic in a go without long division or anything like that.(13 votes) - The equation at5:30i.e. 3^1/2=(a^2+b^2-C^2)/ab can be simplified to (((a+b)^2-c^2)/ab) - 2 by first converting a^2+b^2 to (a+b)^2-2ab and then dividing by ab which can further be simplified to ((a+b-c)(a+b+c)/ab)-2 . Wouldn't this take less time than the lengthy squaring of a,b and c?(7 votes)
- But a^2 + b^2 is not equal to (a+b)^2.
Consider a=3 and b=4...
a^2 + b^2 = 3^2 + 4^2 = 9 + 16 = 25 <> (3+4)^2(0 votes)
- how do you solve periodic function? sin2x=-1/2 0<x<3x(2 votes)
- This function is a trigonometric equation and thus can be solved by using properties of trigonometric functions. The equation will be first changed to sin X= sin α (where x is a variable and α is a constant such that : sin X= sin α), then :X= nπ + (-1)ⁿ.α, ... ∀ n ∈ ℤ. (for α take the principal value and solve)
In the case above sin 2x= sin (-π/6) and the solutions are -π/12 , 5π/12 ... and so on.(here X=2x)
Remember that for such periodic functions (sin x , cos x, tan x)we have several solutions for one value. For each function there is a different result .(2 votes)
- A very easy approach would be use law of cosines and then law of sines too. Using that we will get that a=sqrt(3). And then substitute the value of a in law cosines' equation to get b=c. Which implies that B=C. So x^2-1=2x+1 and solve for x.(2 votes)
- how to remember all the formulaes in trigonometry(1 vote)
- Well firstly if you are aiming for cracking the exam i would recommend . Not to dwell to deep in this topic just going through ncert and some little high level question should be enough . That is because trigonometry questions are being now rarely used in the exam (direct trigonometry questions). Well , now COMING TO YOUR QUESTION . The method that i used was writing down all the formulas and sticking them somewhere you often pass through (study table your room ) I mean anywhere . And whenever you pass through it just give a quick glance for 5 s . believe me it helps a lot you will definitely get it right in 2.5 weeks at most .Hope it helps.(2 votes)
- At6:00, wouldn't it be faster if you just used the formula to do the squares?(1 vote)
- At5:27, why didn't Sal divide each of the terms in the numerator of the R.H.S by "ab" and get a/b - b/a - c^2/ab. Wouldn't it be simpler to perform synthetic division there itself?(1 vote)
- integration of (secx)^3 by ILATE rule if we take first function as (secx)^2 and second function as secx(1 vote)
- I started out working this problem in a different way but i am stuck at one point. Please help.....
I used the sine rule first.. sinA/a = sinB/b=sinC/c=1/2c ( since C = 30 degrees)
so, sinA/a=1/2c
so, a/c= 2sinA ------- I
and similarly b/c = 2sinB-------------II
Adding I and II
a+b/c = 2(sinA+sinB)----------III
a+b/c=x^2+x+1+x^2-1 / 2x+1 = x
so a+b/c = x -----------------IV
from III and IV
x=2(sinA+sinB)
x=2(2sin(A+B/2)cos(A-B/2) )
x=4cos15 cos(A-B/2)
How to find cos(A-B/2) ?(1 vote)- we can find that by using half angle formulas in triangle but that is prolonged process and u will get the same answer as shown in video
sin(A/2)= root of (s-c)(s-b)/cb
cos(A/2)=root of s(s-a)/bc(1 vote)
- but, how do you find minimum and maximum values of trig functions? i know this method:
but some problems like- sqrt(a^2+b^2) greater than/= acos(x) +bsin(x) less than/= sqrt(a^2+b^2)
are hard to do with this method, how do i find their max./min. values?4sin^4(x) + cos^4(x)
(1 vote)
Video transcript
"Let ABC be a triangle,
such that angle ABC is equal to pi over 6." Let me draw it. So that's A, B, and
C. And they tell us that angle ACB, this angle
right over here, is pi over 6. I'm assuming it's
pi over 6 radians. Or we could view
that as 30 degrees. And then they tell
us, "Let a, b, and c denote the lengths of the sides
opposite to capital A, B, C, respectively." So this is a, this is b, and
this is c, right over there. The values of x, for which
a is equal to this thing, b is equal to this thing, and
c is equal to this thing is? So let's think about it. One, we need to find a
relationship between A, B, and C in this angle. And then once we have
that relationship, we can substitute them
with these expressions. And then essentially
try to solve for x. And the first thing you
want to think about is, OK, how can I relate
the length of b, this lowercase c, this lowercase
a with this angle here.? And what should pop
out into your brain is the law of cosines. Law of cosines. Law of cosines that
tells us that c squared is equal to a squared
plus b squared. It's essentially the Pythagorean
theorem with an adjustment right here, because this
isn't a right triangle. So a squared plus b
squared minus 2ab times the cosine of C. Now in this case,
angle C right here-- or I should say, in this case,
it's angle ACB-- this over here is going to be 30 degrees. So what's the cosine
of 30 degrees? And just in case you
forgot it, although I assume if you're going to
take the IIT exam, the JEE, you're going to need all the
help you need in terms of time. So this is 30 degrees. That is 1. Then this is 1/2, the side
opposite the 30-degree side. This will be 60 degrees. It's going to be a 30-60-90. And so this is square
root of 3 over 2. Cosine is adjacent
over hypotenuse, so square root of 3/2 over 1. So this, in this
example, it's going to be c squared is equal
to a squared plus b squared minus 2ab. And then this whole thing
is square root of 3/2. So times square root of 3/2. And these 2's cancel out. And so we get c squared is equal
to a squared plus b squared minus the square root of 3ab. Now the next thing I'd want
to do is substitute for a, I'd substitute with this
expression over here. For b, I want to substitute
with this expression over here. And then for c, I want to
substitute this expression right over here. But I want to be careful,
I want to make sure that it's in a form that I
can, in some way, simplify. Because if I just take this
a over here-- this x squared plus x plus 1-- if
I square it, I'm going to end up with
something to the fourth power. This b, I'm going to end up with
something to the fourth power. And then I have the square
root of 3 business over here. So let's see if we can at least
rearrange it, or at least maybe quarantine the square
root of 3, little bit, to make it at least
something that we might have some hopes of factoring,
or solving in some way. And I want to make a little
clarifying point here. In most Western exams,
if you're taking the SAT or you're taking any type of--
even at a math competition-- if you find yourself
grinding through math, multiplying huge polynomials
times each other, you're probably doing
something wrong. Most Western exams--
like the SAT-- you kind of have
to see the trick, you kind of have to see the
elegance in the problem. But once you see the
elegance, the problem is usually pretty
straightforward. In fact, if you're
grinding through math, it's usually a giveaway that
you're doing something wrong. On these JEE math
problems-- and I've only recently gotten
exposure to them. I obviously never took them,
not having grown up in India. But on these, you have
to kind of see the trick. And then you have to
grind through math, and you have to use every
trick at your disposal to make the grinding through
math a little bit easier. So the test writers for the
JEE are definitely a little, actually, a lot more
cruel than the test writers for most Western
exams that I've seen. So what I want to
do is, let's just isolate the square root of 3ab. So let's add it to both sides. So the square root
of 3ab plus-- so let's just add that--
square root of 3ab. And let's get the c squared
onto this side of the equation. So let me subtract c
squared from both sides of the equation. So minus c squared. And then minus c squared. And so I will have, on this
side, those cancel out. And I just have the
square root of 3ab is equal to a squared plus b
squared-- these cancel out-- minus c squared--
I'm just rearranging the whole thing--
minus c squared. All of that, well right
now, that's all it is. And now I can divide
both sides of this by ab. So if I divide
that by ab and then I divide the right-hand side
by ab, that cancels out. And I've kind of quarantined
that square root of 3. So I'll hopefully only
have integer stuff on this side of the equation. So I have the
square root of 3 is equal to a squared plus b
squared minus c squared. So let's figure out what
each of these things are. Let's expand them out,
actually, before we rewrite it. So let's figure out
what a squared is. So let's think about a squared. a squared is going
to be equal to-- and this is the part
where they clearly want to see whether you can
multiply multiple polynomials. That's going to be x squared
plus x plus 1 squared, or x squared plus x plus 1
times x squared plus x plus 1. And so this is going to be
1 times this whole thing, is x squared plus x plus 1. x times this whole
thing, is x plus x squared plus x to the third. And you may or might
not have noticed, I'm doing these a little faster
than I traditionally do them, just in the hope of
being able to finish this problem in a
reasonable amount of time. And then finally, x squared
times this whole thing, x squared times 1 is x squared. x squared times x
is x to the third. And then x squared times x
squared is x to the fourth. So a squared is going to
be x to the fourth plus 2x to the third plus 3x
squared plus 2x plus 1. So that's a squared. What's b squared? That looks a little bit
more straightforward. b squared, so that's b. So b squared is going to be--
well, that squared is just x to the fourth minus
2x squared plus 1. And then c squared. That's more straightforward. So at least they
weren't as painful as this. c squared is going to
4x squared plus 4x-- 2 times the product of these
two, so plus 4x-- plus 1. So the numerator of
our expression here. So let's rewrite this thing
that we were simplifying. It now becomes the square
root of 3 is equal to-- oh, and we don't want
to-- well, let's just do one step at a time. So it's equal to a squared,
which is this thing over here. So it is x to the
fourth-- actually, let me scroll down a little bit. It's this thing plus this thing,
minus this thing over here. So let me just do it. It's this thing
plus the blue thing, plus x to the fourth
minus 2x squared-- it'll be easier to add
them up this way-- plus 1. And then it's minus this
thing, minus c squared. So minus. So this is going to be a
minus 4x squared minus 4x. And then you have a minus 1. And then when you
add them all up-- so this will be the numerator
of this expression. We have the square root
of 3 is equal to-- so let me just do it down here. So we have 2x to the
fourth-- let's scroll to the left a little bit. So we have 2x to the fourth
plus 2x to the third. And then 3x squared minus 6x
squared is minus 3x squared. 2 minus 4 is minus 2x, and
then you have 2 minus 1 is-- then you have just plus 1. So the numerator, this thing
over here, simplifies to this. It's equal to 2x to the fourth
plus 2x to the third minus 3x squared minus 2x plus 1. And the denominator is ab. a is this thing over here. a is x squared plus x plus 1. And b, we saw it up
here, b is-- actually, we have it written up over
here. b is x squared minus 1. So b is x squared minus 1. Now at this point, we
need to solve for x. If we just multiply this
stuff times square root of 3 and went backwards, kind of
unsolved it back to the form that we had way up here, we'd
have a fourth degree polynomial with square roots of 3's in it. We still have a fourth
degree polynomial. But at least this one has
all integer coefficients, which at least
gives me some hope that I might be
able to factor it. I might be able to
find some roots. And if I can find
some roots that are common to the numerator
and the denominator, maybe I can reduce this
into a lower degree problem. So I'm going to leave you there. I'm pushing 10 minutes. In the next video, we're going
to try to factor this beast up here and see if it has any
common factors with this over here. And then try to solve for x.