How to find tan(x/2)

Question

If sin x=√2/2, and 90° < x < 180, what is tan(x/2)

Answer 1

I completely agree with the above, however, I just wanted to show another formula that might make your life a bit easier.

tan(x/2) = sinx/(1 + cosx)

Since we were given that sinx = √2/2 and 90°< x < 180°, then cosx = -√2/2 (since we're in Q2)

Now tan(x/2) = sinx/(1 + cosx)

= (√2/2)/(1 + √2/2) <-------To simplify, multiply top and bottom by 2

= √2/(2 + √2) <----------Multiply by the conjugate to simplify; thus, multiply by (2 - √2)/(2 - √2)

= √2(2 - √2)/(4 - 2)

= √2(2 - √2)/2

= (2√2 - 2)/2

The nasty part about the formula in the other solution is that you have a ± system to deal with, thus, you need to figure out if your answer is positive or negative first. In this case, you don't have to do that because the formula will just tell you if your answer is positive or negative.

Hope this helps

Mr. K

Answer 2

So the angle is 135.

tan(135/2) = tan ( 67.5) = 2.4142135623730950488016887242097....

Answer 3

There are half-angle formulas. The one for tangent is:

tan(x/2) = ±√(1-cosx)/√(1+cosx)

Given that sin x = √2/2, and 90<x<180, this means that the angle, x, is 135 degrees. And it is in the 2nd quadrant. If take 135/2 we find that x/2 = 67.5 degrees so x/2 is in the 1st quadrant.

The tan(x/2) is either positive or negative, and knowing that x/2 is in the first quadrant, we now know to use the PLUS outside of the radicals.

In the 2nd quadrant, if the sinx = √2/2, then cosx=–√2/2

1 - cosx = 1 - (-√2/2) = 1 + √2/2 using a common denominator this becomes

1 - cosx = (2 + √2)/2

1 + cosx = 1 + (-√2/2) = 1 - √2/2 using a common denominator, this becomes

1 + cosx = (2 - √2)/2

Putting these into the formula for tan(x/2) = √(2+√2) / √(2 - √2)