Half Angle Formula


To find the half-angle formula or t formula as commonly known, we set t = . Afterwards, we use the double angle formula and the trig ratios to obtain the following results for the half angle formula.

= trig function to show the half angle formula


Drawing the triangle to reflect trig ratios,

diagram showing the trig ratios for half angle formula

By comparing the trigonometry ratios, we get the following results.


(3) formula for the half angle of the tangent function


a) Prove that and deduce a similar expression for sec x tan x .Hence find in surd form the value of and .

Let . Remember to always place this before we start solving the problem. Otherwise, we are unable to use the half-angle formula.

Substitute the trigonometry functions with t values.

sec x + tan x =

Then , we do some cancellation as is equal to (1+t)(1-t) .



= (proven)

Working backwards, we are able to prove the trigonometry identity. This requires you to be familiar with the addition formula.


To obtain , we have to find out the value of x.


With the value of x and the result that we have obtained earlier, we can find the surd form for tan .

= result of the half angle formula


Negative angle is at play here. tan x = tan (- x)

We have to adjust it to the form tan

x =

Then we place it into our earlier derived result.



Half Angle Formula in substitution for integration of or

We can also use the half angle formula to solve equations of the form where a,b,c are constant. However, this method should only be used when it is stated. Otherwise, R Formula would be a better option.

Use the half angle formula to solve for values of between 0 and



half angle formula in integration problems

= 1.966,8.249 (not applicable since it falls outside the range)

Yet = is also a solution. This implies that some solutions are missing. Hence when t-formula is used, we have to check whether are solutions.We can do this by substituting into the equation.

Return to Trigonometry Help or Basic Trigonometry .