# 1.06 Transposition of equations involving functions.

## Inverse functions.

Inverse functions perform the opposite actions of their corresponding function.

The Sin function is used on angles to give the sine value of that angle and the Sin^{-1} function is used on sine values, to give the angle
corresponding to that sine value. e.g .

Sin 30^{o} = 0.5 .

Sin^{-1} 0.5 = 30^{o} .

Squaring a term means multiplying it by itself. Square rooting a term, finds the number that when squared, has the same value as the term .

4^{2} = 16 .

16^{1/2} = 4 .

(Note in the last example 16^{1/2} is the same as the square root of 16).

When the term we want to make the subject of the equation is part of a function, first we need to
get the function on its own on one side of the equation, (and not on the bottom line of a fraction).
Then we apply the inverse function, to both sides of the equation (which cancels out the function on one side of the equation).
(Note,we then may require further transposition to get the subject on its own).

**examples**.

- y = Sin θ make θ the subject.
- Sin
^{-1}(y) = θ .
- θ =Sin
^{-1}(y) .

- y = Cos(ωt) make ω the subject.
- Cos
^{-1}(y) = ωt .
- (Cos
^{-1}(y))/t = ω .
- ω = (Cos
^{-1}(y))/y .

- y = (x + z)
^{2} make z the subject .
- y
^{1/2} = x + z .
- y
^{1/2} - x = z .
- z = y
^{1/2} - x .