A Nice Problem of Trigonometry | 2 Ways to Solve

[@PS-mh8ts]

Another method that uses standard formuae for sin(3x) and cos(3x) (The first method presented by MathBooster is the simplest, and this one, just for fun :-)) cos(3x)=4cos³(x)-3cos(x) Hence cos(3x)/cos(x)=4cos²x-3 -- (i) sin(3x)=3sin(x)-4sin³(x) Hence sin(3x)/sin(x)=3-4sin²(x) -- (ii) (i)-(ii) gives: cos(3x)/cos(x)-sin(3x)/sin(x)=4cos²(x)-3-(3-4sin²(x)) =4(cos²(x)+sin²(x))-6=4-6=-2

[@ABHISHEKKUMAR-01024]

We have, cos3x = cos(x + 2x) = cosxcos2x - sinxsin2x = (cosx)(2cos²x - 1) - (sinx)(2sinxcosx) = (2cos³x - cosx) - 2sin²xcosx = 2cos³x - cosx - 2(1 - cos²x)cosx = 2cos³x - cosx - 2(cosx - cos³x) = 2cos³x - cosx - 2cosx + 2cos³x = 4cos³x - 3cosx Also, sin3x = 3sinx - 4sin³x Therefore, (cos3x / cosx) - (sin3x / sinx) = [ (4cos³x - 3cosx) / cosx ] - [ (3sinx - 4sin³x) / sinx ] = [ (cosx)(4cos²x - 3) / cosx ] - [ (sinx)(3 - 4sin²x) / sinx ] = (4cos²x - 3) - (3 - 4sin²x) = 4cos²x - 3 - 3 + 4sin²x = (4cos²x + 4sin²x) - 6 = 4 - 6 = - 2

[@WahranRai]

Look at my solution, simpler

[@honestadministrator]

sin ( x - 2 x) /( sin ( x) cos ( x)) = -2

[@WahranRai]

cos3X = cosXcos2X - sinXsin2X and sin3X = cosXsin2X - sinXcos2X and sin2X = 2sinXcosX ---> cos3X / cosX = cos2X - 2sinXsinX = cos2X - 2(sinX)^2 sin3X / sinX = cos2X + 2cosXcosX = cos2X + 2(cosX)^2 cos3X / cosX - sin3X / sinX = - 2(sinX)^2 - 2(cosX)^2 = -2 ((sinX)^2 + (cosX)^2) = -2

[@stellacollector]

In the 2nd method you have only showed that the value of the given expression is -2 when x=30˚. How does that guarantee that the given expression has the same value for other values of x?

[@MathBooster]

In objective type of questions, you can solve by second way (there is no any restriction on the value of x, so we can put any value that is in the domain). Of course, in subjective questions, you can not use 2nd method.

[@ashokdubey8415]

Use in 2nd method: cos3x=4(cos x)^3-3cos x sin3x =3sin x-4(sin x)^3

[@AAZ3000]

I think the 2 method may be used after 1 method as the guarantee that any value of x within the domain leads to the given trigonometric functions expression is being eventually canceled. In other words, we get the same answer for any value of x within a domain

[@sarantis40kalaitzis48]

I red 2 interesting methods in the comments plus your's 1st method.The easiest is Math Boosters one,but all methods are Mathematically(Scientifically) right and nice.

[@vijayannair2316]

Nice

[@giuseppemalaguti435]

d

[@Stevenzhang-ze8ki]

选择题不要搞得那么复杂！直接让x=30度！答案就出来了。等于-2....