Question

Find all solutions of the equation tan(x/2)+cos(x)=1

Answer 1

Try the tangent half-angle formula.

Answer 2

\[\Huge \tan \left(\frac{\theta}{2} \right)= \frac{\sin\theta}{1 + \cos\theta} = \frac{1-\cos\theta}{\sin\theta}\]

Answer 3

I suggest trying out the second version.

Answer 4

how would my equation fit into that?

Answer 5

You have a \(\tan \frac{x}{2}\), so expand it through the identity.

Answer 6

are you given exact or approximate solutions as answer choices?

Answer 7

my options are:
x=0
x=pi/2, pi
x=0, pi/2
x=0, pi/2, pi
x=0, pi

Answer 8

If you're given options, I'd suggest just plugging in the values and seeing which ones work and which don't.

Answer 9

Alright, thanks for giving the answer choices. It helped me see where I made a typo and I was able to fix it.


tan(x/2) + cos(x) = 1


(sin(x))/(1+cos(x)) + cos(x) = 1


sin(x) + cos(x)(1+cos(x)) = 1+cos(x)


sin(x) + cos(x) + cos^2(x) = 1+cos(x)


sin(x) + cos(x) + cos^2(x) - 1 - cos(x) = 0


sin(x) + cos^2(x) - 1 = 0


sin(x) + 1 - sin^2(x) - 1 = 0


sin(x) - sin^2(x) = 0


-sin^2(x) + sin(x) = 0


-1(sin^2(x) - sin(x)) = 0


sin^2(x) - sin(x) = 0


sin(x)(sin(x) - 1) = 0


sin(x) = 0 or sin(x)-1 = 0


sin(x) = 0 or sin(x) = 1


Solve sin(x) = 0:

sin(x) = 0

x = arcsin(0)

x = 0

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Now solve sin(x) = 1

sin(x) = 1

x = arcsin(1)

x = pi/2


So the solutions are x=0, pi/2