Question 967183: Find sin 2x, cos 2x, and tan 2x from the given information.
sec x = 4, x in Quadrant IV
sin 2x =
cos 2x =
tan 2x =
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! you are given that sec x = 4 in quadrant 4.
secant is equal to 1 / cosine, so cos x = 1/4.
cosine is positive in quadrant 4 so this is good so far.
there's an easy way to do this and there's a hard way to do this.
here's the easy way.
cosine x = 1/4 means that x = 75.52248781 degrees.
that would be in quadrant 1.
in quadrant 4, x would be equal to 360 - 75.52248781 degrees which makes x equal to 284.4775122.
2x is therefore equal to 568.9550244 degrees.
you want to get the equivalent number of degrees between 0 and 360 degrees.
subtract 360 from 568.9550244 to get 208.9550244.
that's equal to 2x.
sin 2x is therefore equal to -.4841229183.
cos 2x is therefore equal to -.875
tan 2x is therefore equal to .5532833352.
the angle of x is in the fourth quadrant.
the angle of 2x is in the third quadrant.
that's the easy way.
now we'll do it the hard way.
we should get the same answer or we did it wrong.
you have sec x = 4 and x is in the fourth quadrant.
since sec x is equal to 1 / cos x, then 1 / cos x = 4 which makes cos x = 1/4.
since cosine is adjacent / hypotenuse, then you get adjacent is equal to 1 and hyupotenuse is equal to 4.
use pythagorus to find opposite.
pythagorus says adjacent squared plus opposite squared = hypotenuse squared.
this becomes 1 squared + opposite squared = 4 squared.
this becomes 1 + opposite squared = 16
solve for opposite squared to get opposite squared = 15.
take the square root of both sides of the equation to get opposite = sqrt(15).
you now have:
adjacent = 1
opposite = sqrt(15)
hypotenuse = 4.
from this, you can derive:
sin x = sqrt(15)/4
cos x = 1/4
tan x = sqrt(15)
this would be if the angle was in the first quadrant because all the functions are positive in the first quadrant.
since the angle is in the fourth quadrant, then sine is negative and tangent is negative while cosine is positive.
we therefore get:
sin x = -sqrt(15)/4
cos x = 1/4
tan x = -sqrt(15)
those are the trig functions in the fourth quadrant where our angle x resides.
now you want to find:
sin 2x =
cos 2x =
tan 2x =
the trig identity formulas for these are:
sin(2x) = 2 * sin(x) * cos(x)
cos(2x) = cos^2(x) - sin^2(x)
tan(2x) = 2 * tan(x) / (1 - tan^2(x))
using what we know of sin(x) and cos(x) and tan(x), we can solve these equations.
we know that:
sin x = -sqrt(15)/4
cos x = 1/4
tan x = -sqrt(15)
sin(2x) = 2 * -sqrt(15) / 4 * 1 / 4 = -sqrt(15) / 8 = -.4841229183.
this agrees with what we found the easy way.
cos(2x) = cos^2(x) - sin^2(x) = (1/4)^2 - (-sqrt(15) / 4)^2 = -.875
this also agrees with what we found the easy way.
tan(2x) = 2 * tan(x) / (1 - tan^2(x)) = 2 * -sqrt(15) / (1 - (-sqrt(15))^2) = .5532833352.
this also agrees with what we found the easy way.
note on the easy way.
it's not as easy as you might think.
you have to know exactly what you are doing or you can screw it up easily.
you were probably meant to solve it the hard way which is thorugh the use of the double angle identity formulas.
i use the easy way to test whether my answer agrees with what i got from the identity formulas.
it does in this case so i'm reasonably confident the solution is good.
the easy way also helps to understand exactly what is happening and why these identity formula work.
but, once again, the easy way is not that easy and don't bother trying to learn it if you don't have to.
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