More trig examples
Fun Trig Problem A trig problem involving the quadratic equation.
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- I received a problem from Bradley.
- I don't know his last name.
- I'm assuming it's a he.
- I don't know where he lives.
- But the problem he gave is interesting.
- And I don't think I've covered this before.
- So I think it's worth covering.
- So the problem he gave, if I read his note properly, is
- this: 3 sine squared of x is equal to 1 plus cosine of x.
- So at first cut, this seems like a difficult problem.
- How do I-- You can't solve for x.
- You would have arcsines and the square roots
- and cosines et cetera.
- Et cetera.
- So the way I approached this is-- Any time that if I see a
- cosine x here but then I see a sine squared x here, I start
- thinking of what trig identities are at my disposal.
- And what trig identities involve a sine squared x?
- Well the most basic trig identity, and this comes out
- of the unit circle definition of trig functions, is that
- sine squared x plus cosine squared x is equal to 1.
- And that comes out of the fact that the equation of a circle
- is x squared plus y squared is equal to the radius squared.
- But it's the unit circle.
- It's equal to 1 squared.
- But anyway.
- Hopefully you have this memorized if you've already
- been watching the trig videos.
- So what does sine squared x equal?
- Well let's solve for it.
- So sine squared x is equal to 1 minus cosine squared x, right?
- So we could substitute this term right here with this.
- And what does that get us?
- Well we're just playing around at this point, but at least
- that way, everything is in terms of cosine of x.
- So let's do that.
- Let's substitute.
- So we get 3 times sine squared of x.
- We just showed that sine squared of x is the same thing
- as 1 minus cosine squared of x.
- Is equal to 1 plus cosine of x.
- We can simplify a little bit.
- 3 minus 3 cosine squared of x is equal to 1 plus cosine of x.
- I don't know.
- Just for kicks, let's put everything onto the right
- side of the equation.
- And you'll see it wasn't just for kicks.
- 0-- right?
- I'm just going to --is equal to-- let's put this onto
- the right side --3 cosine squared x.
- And then-- Let's see.
- We have to subtract 3 from this side.
- Well let's just write the cosine x.
- Plus cosine x.
- And then 1 minus 3 is minus 2.
- Let me make sure I didn't make a careless mistake.
- We have negative 3 here.
- We added 3 cosine x-- 3 cosine squared of x
- to both sides, right?
- We subtracted 3 from both sides.
- Minus 2 and this cosine x is this cosine x.
- Now what can we do?
- Well this is where it gets interesting.
- Because this looks an awful lot like a quadratic equation
- except for the fact that instead of having ax squared
- plus bx plus c, we have a cosine squared x.
- So instead of just having an x squared, we have a whole
- cosine of x squared.
- So what do I mean by that?
- Let me make a substitution.
- And then I think it'll all become clear to you.
- Let's make the substitution that a-- and I'm just picking
- the letter a arbitrarily --is equal to cosine of x.
- So if we were to take the cosine x's of this and replace
- them with a, what do we get?
- And I'm just going to switch it around.
- So I want to put the 0 on that side.
- Equal 0.
- So we get 3-- Well cosine squared x.
- That's the same thing as cosine of x squared, right?
- So we get 3a squared plus a minus 2 is equal to 0.
- Well now we have a pure quadratic.
- And we can solve it using the quadratic equation.
- So what's the quadratic equation?
- Let me write it up here.
- Negative b plus or minus the square root of b
- squared minus 4ac.
- All of that over 2a.
- So what are the roots of this equation?
- Well what's minus-- And remember this a is
- different than this a.
- Maybe I shouldn't have used a as a letter.
- But these-- a, b, and c in the quadratic equation
- represent the coefficients.
- So this is a.
- b is 1.
- And c is just minus 2.
- So what are the roots of this?
- So the a's that solve this. a can equal--
- And I know I confused you.
- I could-- Let me actually write it different.
- Let's make this, instead of a is equal to cosine x, let
- me say that-- I don't know.
- Let me pick a good letter that's not involved in
- the-- Let me say d.
- So 3d squared plus d minus is 2.
- So now the a's, b's and c's are definitely the coefficients.
- So the solutions to this are d-- because I didn't want to
- use a, b, or c --d is equal to minus b.
- Well, b is 1.
- Minus 1.
- And if this is completely foreign to you, you should
- review the videos on the quadratic equation.
- Minus b squared.
- Well that's 1 squared.
- Minus 4ac.
- So minus 4 times a, times 3, times c.
- Well c is minus 2, right?
- So we get a-- That minus cancels there.
- And we have a 2 there.
- All of that over 2 times a. a is 3, so we have it over 6.
- So this equals minus 1 plus or minus the square
- root-- What is this?
- 4 times 3 times 2.
- 24 plus 1.
- 25.
- Oh.
- This works out cleanly.
- Over 6.
- So that equals minus 1 plus or minus 5 over 6.
- And so what are the roots?
- The roots are-- What's minus 1 minus 5?
- That's minus 6 over 6.
- So it's minus 1.
- What's the other one?
- Minus 1 plus 5 is 4.
- 4 over 6 is 2/3.
- So the solution is to the equation-- Let me
- clear up some space.
- Hopefully it'll let me clear up some space here.
- Let me see.
- What was I doing?
- Oh.
- Maybe I want to leave-- I can get rid of this.
- You know the identity.
- And you also know the quadratic formula.
- Let's see.
- Actually, let me get rid of this too.
- Clear up a bunch of space.
- I wanted to leave this here because this showed how this
- turned into a quadratic, but instead of having it in terms
- of just a variable, we have it in terms of cosine of x.
- And then we made this d is equal to cosine of x.
- Anyway.
- So the solution to this equation is that quadratic.
- Is d is equal to minus 1 or 2/3, right?
- But, of course, we made the substitution long ago that
- d is equal to cosine of x.
- So the solution to this equation, in terms of x, is the
- solution to this equation.
- Cosine of x is equal to minus 1 or cosine of x is equal to 2/3.
- Well this one's easy, right?
- x is equal to arccosine of minus 1.
- I always forget if there's two c's when you do arccosine.
- Anyway, so what-- At what degree or radian value does the
- cosine of x equal minus 1?
- Well it's at pi, right?
- So x could equal pi, which is also or 180 degrees.
- This one is not as easy.
- I think I will have to use a calculator for this.
- Unless I'm-- whoops.
- So you may not realize it, but Google is actually
- a calculator.
- And a far more advanced calculator than most.
- So we could use Google to figure out the
- arccosine of 2/3.
- Let's do that.
- Arccosine-- and I don't know if I'm spelling it
- right --of 2 over 3.
- Google tells us that it's 0.841 and a bunch of numbers.
- So x is equal to arccosine of 2 over 3.
- So x is equal to 0.84106.
- Let's see if they work.
- Let's, just for fun, let's just see if this one works.
- Let's see if we substitute pi into this equation we
- get the correct answer.
- Well what's sine of pi?
- Let me erase all of this is so we can check it.
- I'm only going to check pi.
- The 0.84.
- And I don't know.
- That's messy.
- But you could do that in your own time.
- So let's check pi.
- x equals-- No.
- That's not what I wanted to do.
- So what is-- Let's make sure this works with pi.
- 3 sine squared of pi is equal to 1 plus cosine of pi.
- Well what's sine of pi?
- This is equal to 3 sine of pi squared.
- This is equal to 1 plus cosine of pi.
- Well sine of pi is 0, right?
- The y-value when you go 180 degrees is 0.
- So this is 0.
- And what's cosine of pi?
- Cosine of pi is negative 1.
- So 1 plus minus 1.
- Well this is true.
- So pi worked in that equation.
- I think if you substitute that 0841068 whatever, you'd
- also find that that works.
- So thanks Bradley for sending this.
- I thought this was a neat problem because it looks
- like it's trigonometry.
- And it was trigonometry but you had to know a little
- bit of identities.
- And then you had to recognize it as a quadratic equation.
- I will see you in a future video.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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