Half Angle Formula

Concept

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.

=

=

Drawing the triangle to reflect trig ratios,

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

(1)
(2)

(3)

Example

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.

Let

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

=

=

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

Let

then

= 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.