Question

find cos x/2 if tan x = -3/4

Answer 1

Let's ignore the negative sign in -3/4 for now

So if tan x = 3/4, then we have this right triangle

Answer 2

this is because tan(angle) = opposite/adjacent

Answer 3

what is the hypotenuse of the right triangle?

Answer 4

hypotenuse is 5

Answer 5

good

Answer 6

using this triangle, what are sin(x) and cos(x) ?

Answer 7

sin is 3/5 and cos is 4/5

Answer 8

both are correct

Answer 9

To find cos(x/2), we now use the formula


\[\Large \cos\left(\frac{x}{2}\right) = \pm\sqrt{\frac{1+\cos(x)}{2}}\]


So plug in cos(x) = 4/5 and simplify to get...


\[\Large \cos\left(\frac{x}{2}\right) = \pm\sqrt{\frac{1+\cos(x)}{2}}\]
\[\Large \cos\left(\frac{x}{2}\right) = \pm\sqrt{\frac{1+\frac{4}{5}}{2}}\]
\[\Large \cos\left(\frac{x}{2}\right) = \pm\sqrt{\frac{\frac{5}{5}+\frac{4}{5}}{2}}\]
\[\Large \cos\left(\frac{x}{2}\right) = \pm\sqrt{\frac{\frac{5+4}{5}}{2}}\]
\[\Large \cos\left(\frac{x}{2}\right) = \pm\sqrt{\frac{\frac{9}{5}}{2}}\]
\[\Large \cos\left(\frac{x}{2}\right) = \pm\sqrt{\frac{\frac{9}{5}}{\frac{2}{1}}}\]
\[\Large \cos\left(\frac{x}{2}\right) = \pm\sqrt{\frac{9}{5}*\frac{1}{2}}\]
\[\Large \cos\left(\frac{x}{2}\right) = \pm\sqrt{\frac{9}{10}}\]
\[\Large \cos\left(\frac{x}{2}\right) = \pm\frac{\sqrt{9}}{\sqrt{10}}\]
\[\Large \cos\left(\frac{x}{2}\right) = \pm\frac{3}{\sqrt{10}}\]
\[\Large \cos\left(\frac{x}{2}\right) = \pm\frac{3*\sqrt{10}}{\sqrt{10}*\sqrt{10}}\]
\[\Large \cos\left(\frac{x}{2}\right) = \pm\frac{3\sqrt{10}}{10}\]

Answer 10

If you don't have to rationalize the denominator, then you can stop at

\[\Large \cos\left(\frac{x}{2}\right) = \pm\frac{3}{\sqrt{10}}\]

Answer 11

Now the problem is this: tan(x) = -3/4 so x is in quadrant 2 or quadrant 4 (this is where tan(x) is negative)

if x is in quadrant 2, then 90 < x < 180 (degrees)
cut each part in half to get 45 < x/2 < 90

so x/2 is in quadrant 1 if x is in quadrant 2

----------------------------------------

if x is in quadrant 4, then 270 < x < 360 (degrees)
cut each part in half to get 135 < x/2 < 180

so x/2 is in quadrant 2 if x is in quadrant 4

----------------------------------------

so x/2 is either in quadrant 1 or quadrant 2

that means cos(x/2) is either positive or negative

we don't have enough info to say one way or the other

Answer 12

so that's why we are forced to keep the plus/minus (unless more info comes along)

Answer 13

cool, thank you

Answer 14

you're welcome